| L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s − 3·9-s + 10-s − 11-s + 2·13-s − 16-s + 6·17-s − 3·18-s − 4·19-s − 20-s − 22-s + 4·23-s + 25-s + 2·26-s + 6·29-s − 8·31-s + 5·32-s + 6·34-s + 3·36-s − 2·37-s − 4·38-s − 3·40-s + 2·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s − 9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 1.43·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s − 0.474·40-s + 0.312·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.028669856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028669856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.87836070551548265134861946988, −14.22243129432120604396205611002, −13.19684320404801561766277475369, −12.23057998502845892155732184830, −10.82193865307292862841462120688, −9.365267708430525132762162914792, −8.248642768475844318802620430882, −6.16821463973303244938343944630, −5.09766059609940450216711100862, −3.25423179226253980082861279896,
3.25423179226253980082861279896, 5.09766059609940450216711100862, 6.16821463973303244938343944630, 8.248642768475844318802620430882, 9.365267708430525132762162914792, 10.82193865307292862841462120688, 12.23057998502845892155732184830, 13.19684320404801561766277475369, 14.22243129432120604396205611002, 14.87836070551548265134861946988
