| L(s) = 1 | − 16·4-s + 30·5-s − 36·9-s − 240·11-s + 160·16-s + 152·19-s − 480·20-s + 656·25-s − 396·29-s − 1.14e3·31-s + 576·36-s − 168·41-s + 3.84e3·44-s − 1.08e3·45-s + 1.44e3·49-s − 7.20e3·55-s − 840·59-s − 1.38e3·61-s − 1.28e3·64-s − 828·71-s − 2.43e3·76-s + 740·79-s + 4.80e3·80-s + 810·81-s + 180·89-s + 4.56e3·95-s + 8.64e3·99-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2.68·5-s − 4/3·9-s − 6.57·11-s + 5/2·16-s + 1.83·19-s − 5.36·20-s + 5.24·25-s − 2.53·29-s − 6.65·31-s + 8/3·36-s − 0.639·41-s + 13.1·44-s − 3.57·45-s + 4.19·49-s − 17.6·55-s − 1.85·59-s − 2.90·61-s − 5/2·64-s − 1.38·71-s − 3.67·76-s + 1.05·79-s + 6.70·80-s + 10/9·81-s + 0.214·89-s + 4.92·95-s + 8.77·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4150282025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4150282025\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 5 | \( 1 - 6 p T + 244 T^{2} + 678 p T^{3} - 3258 p^{2} T^{4} + 678 p^{4} T^{5} + 244 p^{6} T^{6} - 6 p^{10} T^{7} + p^{12} T^{8} \) |
| 19 | \( ( 1 - p T )^{8} \) |
| good | 7 | \( 1 - 1440 T^{2} + 1130344 T^{4} - 87356340 p T^{6} + 243297389910 T^{8} - 87356340 p^{7} T^{10} + 1130344 p^{12} T^{12} - 1440 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 + 120 T + 9472 T^{2} + 46362 p T^{3} + 178230 p^{2} T^{4} + 46362 p^{4} T^{5} + 9472 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 5820 T^{2} + 14612272 T^{4} - 19273004928 T^{6} + 22663961073894 T^{8} - 19273004928 p^{6} T^{10} + 14612272 p^{12} T^{12} - 5820 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 29868 T^{2} + 416686084 T^{4} - 3587191558596 T^{6} + 21050893068184470 T^{8} - 3587191558596 p^{6} T^{10} + 416686084 p^{12} T^{12} - 29868 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( 1 - 32856 T^{2} + 799017388 T^{4} - 12253039281432 T^{6} + 171260630639169222 T^{8} - 12253039281432 p^{6} T^{10} + 799017388 p^{12} T^{12} - 32856 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 + 198 T + 3152 p T^{2} + 13425972 T^{3} + 3248092362 T^{4} + 13425972 p^{3} T^{5} + 3152 p^{7} T^{6} + 198 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 574 T + 218380 T^{2} + 55964578 T^{3} + 11172116974 T^{4} + 55964578 p^{3} T^{5} + 218380 p^{6} T^{6} + 574 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 155432 T^{2} + 16573328416 T^{4} - 1254563573864564 T^{6} + 71329690235328334726 T^{8} - 1254563573864564 p^{6} T^{10} + 16573328416 p^{12} T^{12} - 155432 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 + 84 T + 40988 T^{2} - 26143758 T^{3} - 1743639810 T^{4} - 26143758 p^{3} T^{5} + 40988 p^{6} T^{6} + 84 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 410960 T^{2} + 81635095504 T^{4} - 10511382711487340 T^{6} + \)\(97\!\cdots\!90\)\( T^{8} - 10511382711487340 p^{6} T^{10} + 81635095504 p^{12} T^{12} - 410960 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 433816 T^{2} + 105467558860 T^{4} - 17252874982537432 T^{6} + \)\(20\!\cdots\!98\)\( T^{8} - 17252874982537432 p^{6} T^{10} + 105467558860 p^{12} T^{12} - 433816 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 823564 T^{2} + 330771109060 T^{4} - 84209316042232708 T^{6} + \)\(14\!\cdots\!98\)\( T^{8} - 84209316042232708 p^{6} T^{10} + 330771109060 p^{12} T^{12} - 823564 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 + 420 T + 706492 T^{2} + 249270228 T^{3} + 206609980614 T^{4} + 249270228 p^{3} T^{5} + 706492 p^{6} T^{6} + 420 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 692 T + 213148 T^{2} - 134075176 T^{3} - 107383307330 T^{4} - 134075176 p^{3} T^{5} + 213148 p^{6} T^{6} + 692 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 10148 p T^{2} + 405569140372 T^{4} - 181320682000918772 T^{6} + \)\(56\!\cdots\!14\)\( T^{8} - 181320682000918772 p^{6} T^{10} + 405569140372 p^{12} T^{12} - 10148 p^{19} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 + 414 T + 350224 T^{2} + 68843838 T^{3} + 73704587166 T^{4} + 68843838 p^{3} T^{5} + 350224 p^{6} T^{6} + 414 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 1844444 T^{2} + 1844392383172 T^{4} - 1195444690166044388 T^{6} + \)\(54\!\cdots\!74\)\( T^{8} - 1195444690166044388 p^{6} T^{10} + 1844392383172 p^{12} T^{12} - 1844444 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 370 T + 519436 T^{2} + 4764842 p T^{3} - 2319674 p^{2} T^{4} + 4764842 p^{4} T^{5} + 519436 p^{6} T^{6} - 370 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 2928772 T^{2} + 4138940004004 T^{4} - 3832111525070898124 T^{6} + \)\(25\!\cdots\!30\)\( T^{8} - 3832111525070898124 p^{6} T^{10} + 4138940004004 p^{12} T^{12} - 2928772 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 90 T + 1210732 T^{2} - 94595772 T^{3} + 1337364121014 T^{4} - 94595772 p^{3} T^{5} + 1210732 p^{6} T^{6} - 90 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 5159124 T^{2} + 12667669247488 T^{4} - 19567082172382547688 T^{6} + \)\(22\!\cdots\!58\)\( p^{2} T^{8} - 19567082172382547688 p^{6} T^{10} + 12667669247488 p^{12} T^{12} - 5159124 p^{18} T^{14} + p^{24} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.31817773035118681299798039457, −4.09653192067403442738442748363, −3.80832670613092818021726306402, −3.77714610875286874376391088475, −3.60559665635345780396085762395, −3.59944955570795890082800795347, −3.26347717586484786989727836810, −3.06202818123865253891476758663, −3.01195852156469071745625235883, −2.76377366561680192541311710759, −2.71255929056716236137470804932, −2.67495180744806300553363953054, −2.51822713107485575139758291790, −2.48742940802745916421288236589, −1.92650053572646859530672028023, −1.86585424634480340867545575151, −1.83146259545813731440489917612, −1.64788504889709147118430871101, −1.43373112292331452673515953707, −1.42658331181927314890744132910, −0.65621351933430835568296368920, −0.49239396741110074287568566534, −0.44324283188350804122041298712, −0.39826934861358808839674390172, −0.092679640404717781888345501996,
0.092679640404717781888345501996, 0.39826934861358808839674390172, 0.44324283188350804122041298712, 0.49239396741110074287568566534, 0.65621351933430835568296368920, 1.42658331181927314890744132910, 1.43373112292331452673515953707, 1.64788504889709147118430871101, 1.83146259545813731440489917612, 1.86585424634480340867545575151, 1.92650053572646859530672028023, 2.48742940802745916421288236589, 2.51822713107485575139758291790, 2.67495180744806300553363953054, 2.71255929056716236137470804932, 2.76377366561680192541311710759, 3.01195852156469071745625235883, 3.06202818123865253891476758663, 3.26347717586484786989727836810, 3.59944955570795890082800795347, 3.60559665635345780396085762395, 3.77714610875286874376391088475, 3.80832670613092818021726306402, 4.09653192067403442738442748363, 4.31817773035118681299798039457
Plot not available for L-functions of degree greater than 10.
