| L(s) = 1 | − 2-s + 6·3-s + 9·4-s − 6·6-s − 24·7-s − 25·8-s + 27·9-s − 22·11-s + 54·12-s − 30·13-s + 24·14-s + 41·16-s − 106·17-s − 27·18-s + 50·19-s − 144·21-s + 22·22-s − 134·23-s − 150·24-s + 30·26-s + 108·27-s − 216·28-s − 198·29-s + 360·31-s − 249·32-s − 132·33-s + 106·34-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s + 9/8·4-s − 0.408·6-s − 1.29·7-s − 1.10·8-s + 9-s − 0.603·11-s + 1.29·12-s − 0.640·13-s + 0.458·14-s + 0.640·16-s − 1.51·17-s − 0.353·18-s + 0.603·19-s − 1.49·21-s + 0.213·22-s − 1.21·23-s − 1.27·24-s + 0.226·26-s + 0.769·27-s − 1.45·28-s − 1.26·29-s + 2.08·31-s − 1.37·32-s − 0.696·33-s + 0.534·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T - p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 24 T + 442 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 30 T + 4522 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 106 T + 7882 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 50 T + 14246 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 134 T + 26398 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 198 T + 57706 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 328 T + 62630 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 386 T + 179870 T^{2} + 386 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 266 T + 92542 T^{2} + 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 522 T + 295162 T^{2} - 522 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 172 T + 175654 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 776 T + 528582 T^{2} - 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 630 T + 744334 T^{2} - 630 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1296 T + 1178926 T^{2} + 1296 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 652 T + 589506 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 324 T + 579670 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 756 T + 1427110 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 452 T + 982470 T^{2} - 452 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.505782475965799818981688024496, −9.469347718018433822643077355653, −8.617583193755793858561833080536, −8.442709499838904485943608224778, −8.047811186628844589726066867079, −7.31502066530146943600225182901, −7.21961937914487229477665703058, −6.50080582693094776380446550668, −6.42431338613197450313024015287, −5.95465640471383046372825201340, −5.09406308198857830745715990997, −4.68581540727749609673383706107, −3.87565372070342882810123320050, −3.35637830853736549548861610668, −3.02134966522671574925917699623, −2.30985732823011597579409948220, −2.27134909059286964123138335012, −1.37992906861828095508025257151, 0, 0,
1.37992906861828095508025257151, 2.27134909059286964123138335012, 2.30985732823011597579409948220, 3.02134966522671574925917699623, 3.35637830853736549548861610668, 3.87565372070342882810123320050, 4.68581540727749609673383706107, 5.09406308198857830745715990997, 5.95465640471383046372825201340, 6.42431338613197450313024015287, 6.50080582693094776380446550668, 7.21961937914487229477665703058, 7.31502066530146943600225182901, 8.047811186628844589726066867079, 8.442709499838904485943608224778, 8.617583193755793858561833080536, 9.469347718018433822643077355653, 9.505782475965799818981688024496
