| L(s) = 1 | + 2·3-s + 5·9-s + 6·17-s − 4·23-s + 22·25-s + 2·27-s + 4·29-s − 30·43-s − 7·49-s + 12·51-s + 32·53-s − 28·61-s − 8·69-s + 44·75-s − 56·79-s + 12·81-s + 8·87-s − 4·101-s − 8·103-s + 16·107-s − 40·113-s − 8·121-s + 127-s − 60·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 5/3·9-s + 1.45·17-s − 0.834·23-s + 22/5·25-s + 0.384·27-s + 0.742·29-s − 4.57·43-s − 49-s + 1.68·51-s + 4.39·53-s − 3.58·61-s − 0.963·69-s + 5.08·75-s − 6.30·79-s + 4/3·81-s + 0.857·87-s − 0.398·101-s − 0.788·103-s + 1.54·107-s − 3.76·113-s − 0.727·121-s + 0.0887·127-s − 5.28·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.406083045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.406083045\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 - T - T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 - 11 T^{2} + 76 T^{4} - 11 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + p T^{2} + 45 T^{4} - 94 p T^{6} - 4786 T^{8} - 94 p^{3} T^{10} + 45 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 8 T^{2} - 126 T^{4} - 416 T^{6} + 13715 T^{8} - 416 p^{2} T^{10} - 126 p^{4} T^{12} + 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 3 T - 23 T^{2} + 6 T^{3} + 582 T^{4} + 6 p T^{5} - 23 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 2 T - 26 T^{2} - 32 T^{3} + 279 T^{4} - 32 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 2 T - 38 T^{2} + 32 T^{3} + 807 T^{4} + 32 p T^{5} - 38 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 + 27 T^{2} - 1235 T^{4} - 20898 T^{6} + 985134 T^{8} - 20898 p^{2} T^{10} - 1235 p^{4} T^{12} + 27 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 15 T + 87 T^{2} + 780 T^{3} + 7520 T^{4} + 780 p T^{5} + 87 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 119 T^{2} + 7444 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 18 T^{2} - 3157 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 14 T + 42 T^{2} + 448 T^{3} + 8039 T^{4} + 448 p T^{5} + 42 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 184 T^{2} + 18114 T^{4} + 1244576 T^{6} + 75679859 T^{8} + 1244576 p^{2} T^{10} + 18114 p^{4} T^{12} + 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 + 207 T^{2} + 22093 T^{4} + 2209518 T^{6} + 189398046 T^{8} + 2209518 p^{2} T^{10} + 22093 p^{4} T^{12} + 207 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 32 T^{2} + 5406 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 14 T + 190 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( 1 + 320 T^{2} + 61026 T^{4} + 8170240 T^{6} + 837955235 T^{8} + 8170240 p^{2} T^{10} + 61026 p^{4} T^{12} + 320 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 + 244 T^{2} + 26922 T^{4} + 3366224 T^{6} + 414969491 T^{8} + 3366224 p^{2} T^{10} + 26922 p^{4} T^{12} + 244 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.08952509653937152134061955993, −3.94013526863016458013794836581, −3.93904284862228201693412775433, −3.57054538557179629371329663766, −3.53127990691954008946291818592, −3.40963919426020332890239155885, −3.19404131311396173445731472118, −3.15388815480846979245738274678, −3.13886480293255122480943622218, −2.97184473145789137261748530825, −2.70539399801345273894665110926, −2.59975243692072302994116280196, −2.56354334878589187149051336808, −2.50353783307600571731960929160, −2.25972310460444782255059778981, −2.01947867417727696228173797228, −1.67353972291194256943318519271, −1.59598784583723879451999404670, −1.50422825840110135847922443859, −1.38182971271718466068493757625, −1.31600202629898155071560550164, −1.19569140318291721063217536213, −0.68603763325411202769176445239, −0.56251588684441340830057399597, −0.20006475623349216494884760485,
0.20006475623349216494884760485, 0.56251588684441340830057399597, 0.68603763325411202769176445239, 1.19569140318291721063217536213, 1.31600202629898155071560550164, 1.38182971271718466068493757625, 1.50422825840110135847922443859, 1.59598784583723879451999404670, 1.67353972291194256943318519271, 2.01947867417727696228173797228, 2.25972310460444782255059778981, 2.50353783307600571731960929160, 2.56354334878589187149051336808, 2.59975243692072302994116280196, 2.70539399801345273894665110926, 2.97184473145789137261748530825, 3.13886480293255122480943622218, 3.15388815480846979245738274678, 3.19404131311396173445731472118, 3.40963919426020332890239155885, 3.53127990691954008946291818592, 3.57054538557179629371329663766, 3.93904284862228201693412775433, 3.94013526863016458013794836581, 4.08952509653937152134061955993
Plot not available for L-functions of degree greater than 10.
