| L(s) = 1 | − 9·2-s − 6·3-s + 11·4-s − 18·5-s + 54·6-s − 98·7-s + 243·8-s + 54·9-s + 162·10-s − 66·12-s + 350·13-s + 882·14-s + 108·15-s − 1.38e3·16-s − 1.80e3·17-s − 486·18-s + 3.26e3·19-s − 198·20-s + 588·21-s + 2.08e3·23-s − 1.45e3·24-s − 4.58e3·25-s − 3.15e3·26-s − 1.89e3·27-s − 1.07e3·28-s − 6.69e3·29-s − 972·30-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.384·3-s + 0.343·4-s − 0.321·5-s + 0.612·6-s − 0.755·7-s + 1.34·8-s + 2/9·9-s + 0.512·10-s − 0.132·12-s + 0.574·13-s + 1.20·14-s + 0.123·15-s − 1.35·16-s − 1.51·17-s − 0.353·18-s + 2.07·19-s − 0.110·20-s + 0.290·21-s + 0.823·23-s − 0.516·24-s − 1.46·25-s − 0.913·26-s − 0.498·27-s − 0.259·28-s − 1.47·29-s − 0.197·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7268442108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7268442108\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 9 T + 35 p T^{2} + 9 p^{5} T^{3} + p^{10} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 p T - 2 p^{2} T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 18 T + 4906 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 350 T + 546978 T^{2} - 350 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1800 T + 3567406 T^{2} + 1800 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3266 T + 7614270 T^{2} - 3266 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2088 T + 9365230 T^{2} - 2088 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6696 T + 51326470 T^{2} + 6696 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 20 T + 53103102 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6048 T + 223864366 T^{2} - 6048 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3020 T - 30383466 T^{2} - 3020 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11700 T + 292735582 T^{2} - 11700 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9468 T + 858185230 T^{2} - 9468 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 43938 T + 1852599934 T^{2} + 43938 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 64754 T + 2408321418 T^{2} - 64754 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 24784 T + 2799959190 T^{2} - 24784 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17452 T + 3828622374 T^{2} + 17452 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 51256 T + 3645565854 T^{2} + 51256 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 117558 T + 7798161502 T^{2} + 117558 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 84276 T + 5915697430 T^{2} - 84276 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.630426703503209175186660642980, −9.267844495615412876094120599045, −8.905263828384412280432038195909, −8.547577301990521883958323233615, −8.066572470268800350162078373723, −7.51872442025215991272682462136, −7.22684309546307051823544621330, −7.06007572286584870636508127462, −6.06871599376353554537243636984, −5.95398862695451795222335247742, −5.36173401114138768389962416067, −4.72775627617332148058120873920, −4.29543675493320155443295200438, −3.66557485530379533053202569198, −3.41402249701064925120982412185, −2.47935009784096931257148743634, −1.91772140170548664357819384123, −1.21530999129728489875068560683, −0.53454319265707311897438087908, −0.49526397864525322482562300664,
0.49526397864525322482562300664, 0.53454319265707311897438087908, 1.21530999129728489875068560683, 1.91772140170548664357819384123, 2.47935009784096931257148743634, 3.41402249701064925120982412185, 3.66557485530379533053202569198, 4.29543675493320155443295200438, 4.72775627617332148058120873920, 5.36173401114138768389962416067, 5.95398862695451795222335247742, 6.06871599376353554537243636984, 7.06007572286584870636508127462, 7.22684309546307051823544621330, 7.51872442025215991272682462136, 8.066572470268800350162078373723, 8.547577301990521883958323233615, 8.905263828384412280432038195909, 9.267844495615412876094120599045, 9.630426703503209175186660642980
