| L(s) = 1 | − 20·7-s + 28·13-s − 8·19-s + 10·25-s + 28·31-s − 128·37-s − 116·43-s + 256·49-s + 4·61-s − 56·67-s − 56·73-s + 268·79-s − 560·91-s − 8·97-s + 400·103-s + 280·109-s − 376·121-s + 127-s + 131-s + 160·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2.85·7-s + 2.15·13-s − 0.421·19-s + 2/5·25-s + 0.903·31-s − 3.45·37-s − 2.69·43-s + 5.22·49-s + 4/61·61-s − 0.835·67-s − 0.767·73-s + 3.39·79-s − 6.15·91-s − 0.0824·97-s + 3.88·103-s + 2.56·109-s − 3.10·121-s + 0.00787·127-s + 0.00763·131-s + 1.20·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5836914652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5836914652\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| good | 7 | \( ( 1 + 10 T + 22 T^{2} - 200 T^{3} - 1217 T^{4} - 200 p^{2} T^{5} + 22 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 376 T^{2} + 78370 T^{4} + 12680224 T^{6} + 1685511139 T^{8} + 12680224 p^{4} T^{10} + 78370 p^{8} T^{12} + 376 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 - 14 T - 146 T^{2} - 56 T^{3} + 55279 T^{4} - 56 p^{2} T^{5} - 146 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 353 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 + 2 T + 543 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( 1 + 38 p T^{2} + 330745 T^{4} - 4808938 p T^{6} - 104472345836 T^{8} - 4808938 p^{5} T^{10} + 330745 p^{8} T^{12} + 38 p^{13} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 + 3256 T^{2} + 6538210 T^{4} + 8624375584 T^{6} + 8512106785699 T^{8} + 8624375584 p^{4} T^{10} + 6538210 p^{8} T^{12} + 3256 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 14 T - 1595 T^{2} + 1834 T^{3} + 2095804 T^{4} + 1834 p^{2} T^{5} - 1595 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 + 32 T + 1374 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 + 4312 T^{2} + 8657986 T^{4} + 18471900832 T^{6} + 38186067024259 T^{8} + 18471900832 p^{4} T^{10} + 8657986 p^{8} T^{12} + 4312 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 58 T + 1030 T^{2} - 79112 T^{3} - 4191281 T^{4} - 79112 p^{2} T^{5} + 1030 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 + 8188 T^{2} + 40626826 T^{4} + 136388793328 T^{6} + 346871004689299 T^{8} + 136388793328 p^{4} T^{10} + 40626826 p^{8} T^{12} + 8188 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( ( 1 - 2218 T^{2} + 16848843 T^{4} - 2218 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 7912 T^{2} + 26928706 T^{4} + 90484132192 T^{6} + 367048817837539 T^{8} + 90484132192 p^{4} T^{10} + 26928706 p^{8} T^{12} + 7912 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 2 T - 959 T^{2} + 12958 T^{3} - 12933356 T^{4} + 12958 p^{2} T^{5} - 959 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 14 T - 4293 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 71 | \( ( 1 - 11704 T^{2} + 67208766 T^{4} - 11704 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 + 14 T + 10302 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 134 T + 7465 T^{2} + 266794 T^{3} - 40952396 T^{4} + 266794 p^{2} T^{5} + 7465 p^{4} T^{6} - 134 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 2338 T^{2} - 40635839 T^{4} - 114128438942 T^{6} - 386308937616476 T^{8} - 114128438942 p^{4} T^{10} - 40635839 p^{8} T^{12} + 2338 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 5512 T^{2} + 128866398 T^{4} - 5512 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 4 T - 18626 T^{2} - 704 T^{3} + 258844339 T^{4} - 704 p^{2} T^{5} - 18626 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−3.65883472159432653571517677727, −3.58608286637955001075952130438, −3.55863562477775854908640898053, −3.53578704230017208309227461240, −3.31224737598937148803839875727, −3.13072826809635451213120786783, −2.95879133530021215186369600474, −2.93867480382695305968408676982, −2.91912333784811674514247236884, −2.52002090131086952960280345190, −2.50829603504071682419794884125, −2.41818438558740635271946395241, −2.13353095978684806082180039537, −1.94178855512327625494667991152, −1.88527447057827602977863593004, −1.76703912983876248352806980957, −1.64868193313332572271664339107, −1.27921123532902005630494551857, −1.14639922806613731160237692266, −1.13431174764937880287901526630, −0.913098598141007074683826303175, −0.56414818573800756812641238608, −0.48188617691692294171128384610, −0.30691212912986200593499615800, −0.07320100026357682183324408179,
0.07320100026357682183324408179, 0.30691212912986200593499615800, 0.48188617691692294171128384610, 0.56414818573800756812641238608, 0.913098598141007074683826303175, 1.13431174764937880287901526630, 1.14639922806613731160237692266, 1.27921123532902005630494551857, 1.64868193313332572271664339107, 1.76703912983876248352806980957, 1.88527447057827602977863593004, 1.94178855512327625494667991152, 2.13353095978684806082180039537, 2.41818438558740635271946395241, 2.50829603504071682419794884125, 2.52002090131086952960280345190, 2.91912333784811674514247236884, 2.93867480382695305968408676982, 2.95879133530021215186369600474, 3.13072826809635451213120786783, 3.31224737598937148803839875727, 3.53578704230017208309227461240, 3.55863562477775854908640898053, 3.58608286637955001075952130438, 3.65883472159432653571517677727
Plot not available for L-functions of degree greater than 10.
