| L(s) = 1 | − 260·3-s + 1.25e3·5-s − 1.70e3·7-s + 1.53e4·9-s − 2.39e4·11-s + 1.15e5·13-s − 3.25e5·15-s + 4.12e5·17-s + 2.96e5·19-s + 4.42e5·21-s + 1.04e6·23-s + 1.17e6·25-s + 4.46e6·27-s − 3.66e6·29-s − 1.61e6·31-s + 6.23e6·33-s − 2.12e6·35-s − 2.11e7·37-s − 2.99e7·39-s − 2.69e7·41-s − 5.28e7·43-s + 1.92e7·45-s − 5.84e7·47-s − 3.23e7·49-s − 1.07e8·51-s − 3.90e7·53-s − 2.99e7·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 0.894·5-s − 0.267·7-s + 0.780·9-s − 0.493·11-s + 1.11·13-s − 1.65·15-s + 1.19·17-s + 0.521·19-s + 0.495·21-s + 0.781·23-s + 3/5·25-s + 1.61·27-s − 0.962·29-s − 0.313·31-s + 0.915·33-s − 0.239·35-s − 1.85·37-s − 2.06·39-s − 1.48·41-s − 2.35·43-s + 0.698·45-s − 1.74·47-s − 0.801·49-s − 2.22·51-s − 0.679·53-s − 0.441·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 260 T + 17410 p T^{2} + 260 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1700 T + 5031650 p T^{2} + 1700 p^{9} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 23984 T + 1217213446 T^{2} + 23984 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 115020 T + 22672043710 T^{2} - 115020 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 412820 T + 113016614470 T^{2} - 412820 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 296520 T + 659218232758 T^{2} - 296520 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1049220 T + 3497852029390 T^{2} - 1049220 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3666980 T + 20832571957438 T^{2} + 3666980 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 1613144 T + 29382323902526 T^{2} + 1613144 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 21121940 T + 328931801286510 T^{2} + 21121940 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 26957276 T + 811945448362966 T^{2} + 26957276 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 52889700 T + 1703843788760950 T^{2} + 52889700 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 58412180 T + 2814913257457630 T^{2} + 58412180 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 39035140 T + 5675030678030830 T^{2} + 39035140 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 54995560 T + 15674484224932678 T^{2} - 54995560 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 274579716 T + 41753623519328446 T^{2} + 274579716 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 318580 T + 48520062064444070 T^{2} - 318580 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 7130936 T + 51935375688707086 T^{2} - 7130936 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 120858180 T + 42707263689423190 T^{2} - 120858180 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6877520 T - 115982362290712162 T^{2} + 6877520 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1402348740 T + 857904310704391270 T^{2} + 1402348740 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 830088660 T + 692293619421117718 T^{2} - 830088660 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 638394580 T + 1615411126351062630 T^{2} - 638394580 p^{9} T^{3} + p^{18} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.91633346612184907332023860043, −11.85317479276560552781097868852, −10.96395379460724678797794341208, −10.92536582444603334814427027711, −9.953519134626297116565887974812, −9.933214454921918246245303151545, −8.811155577028848079224379236827, −8.430755846353980198699704466515, −7.47995273412471680657756101568, −6.59874503056738560587136966161, −6.36211497029717350067649745801, −5.64354649690455642131761236638, −5.18501360442173661154752827800, −4.96377256152341887831621215713, −3.37542349387348686684900784821, −3.14381093470579667682176944859, −1.61692099329015682280055185277, −1.29622056076625173988817307107, 0, 0,
1.29622056076625173988817307107, 1.61692099329015682280055185277, 3.14381093470579667682176944859, 3.37542349387348686684900784821, 4.96377256152341887831621215713, 5.18501360442173661154752827800, 5.64354649690455642131761236638, 6.36211497029717350067649745801, 6.59874503056738560587136966161, 7.47995273412471680657756101568, 8.430755846353980198699704466515, 8.811155577028848079224379236827, 9.933214454921918246245303151545, 9.953519134626297116565887974812, 10.92536582444603334814427027711, 10.96395379460724678797794341208, 11.85317479276560552781097868852, 11.91633346612184907332023860043
