| L(s) = 1 | − 10·2-s + 68·4-s + 106·5-s − 49·7-s − 360·8-s − 1.06e3·10-s − 92·11-s + 670·13-s + 490·14-s + 1.42e3·16-s + 222·17-s − 908·19-s + 7.20e3·20-s + 920·22-s + 1.17e3·23-s + 8.11e3·25-s − 6.70e3·26-s − 3.33e3·28-s − 1.11e3·29-s + 3.69e3·31-s − 2.72e3·32-s − 2.22e3·34-s − 5.19e3·35-s + 4.18e3·37-s + 9.08e3·38-s − 3.81e4·40-s + 6.66e3·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s + 1.89·5-s − 0.377·7-s − 1.98·8-s − 3.35·10-s − 0.229·11-s + 1.09·13-s + 0.668·14-s + 1.39·16-s + 0.186·17-s − 0.577·19-s + 4.02·20-s + 0.405·22-s + 0.463·23-s + 2.59·25-s − 1.94·26-s − 0.803·28-s − 0.246·29-s + 0.690·31-s − 0.469·32-s − 0.329·34-s − 0.716·35-s + 0.502·37-s + 1.02·38-s − 3.77·40-s + 0.618·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.051772530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051772530\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 106 T + p^{5} T^{2} \) |
| 11 | \( 1 + 92 T + p^{5} T^{2} \) |
| 13 | \( 1 - 670 T + p^{5} T^{2} \) |
| 17 | \( 1 - 222 T + p^{5} T^{2} \) |
| 19 | \( 1 + 908 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1176 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1118 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3696 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6662 T + p^{5} T^{2} \) |
| 43 | \( 1 + 3700 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7056 T + p^{5} T^{2} \) |
| 53 | \( 1 - 37578 T + p^{5} T^{2} \) |
| 59 | \( 1 + 32700 T + p^{5} T^{2} \) |
| 61 | \( 1 + 10802 T + p^{5} T^{2} \) |
| 67 | \( 1 - 64996 T + p^{5} T^{2} \) |
| 71 | \( 1 - 61320 T + p^{5} T^{2} \) |
| 73 | \( 1 - 38922 T + p^{5} T^{2} \) |
| 79 | \( 1 + 88096 T + p^{5} T^{2} \) |
| 83 | \( 1 + 71892 T + p^{5} T^{2} \) |
| 89 | \( 1 + 111818 T + p^{5} T^{2} \) |
| 97 | \( 1 + 150846 T + p^{5} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.92084141542177293929001006090, −12.85788543879532501420650681769, −11.04076635736202701437140695531, −10.19017225495751400382472419806, −9.381894354856711328006619729723, −8.474968600097594610583662324545, −6.78527269508246588562802604332, −5.84161629605689731879169257754, −2.47502002723649242430446656244, −1.15317495810816317271326362005,
1.15317495810816317271326362005, 2.47502002723649242430446656244, 5.84161629605689731879169257754, 6.78527269508246588562802604332, 8.474968600097594610583662324545, 9.381894354856711328006619729723, 10.19017225495751400382472419806, 11.04076635736202701437140695531, 12.85788543879532501420650681769, 13.92084141542177293929001006090
