| L(s) = 1 | + 2-s − 2.56·3-s + 4-s + 2·5-s − 2.56·6-s − 0.561·7-s + 8-s + 3.56·9-s + 2·10-s + 11-s − 2.56·12-s − 0.561·13-s − 0.561·14-s − 5.12·15-s + 16-s − 0.561·17-s + 3.56·18-s + 2·20-s + 1.43·21-s + 22-s + 1.43·23-s − 2.56·24-s − 25-s − 0.561·26-s − 1.43·27-s − 0.561·28-s + 5.68·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.894·5-s − 1.04·6-s − 0.212·7-s + 0.353·8-s + 1.18·9-s + 0.632·10-s + 0.301·11-s − 0.739·12-s − 0.155·13-s − 0.150·14-s − 1.32·15-s + 0.250·16-s − 0.136·17-s + 0.839·18-s + 0.447·20-s + 0.313·21-s + 0.213·22-s + 0.299·23-s − 0.522·24-s − 0.200·25-s − 0.110·26-s − 0.276·27-s − 0.106·28-s + 1.05·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200341388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200341388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 - 7.68T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 0.876T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−7.49656327108103711519974684412, −6.77258882591034770576810068363, −6.20922550054195113709633633278, −5.87130281168689492098727740929, −5.05917114168770782977842868853, −4.64430177490141412929841993326, −3.68183590469577613779774575966, −2.66834210528100327403904088613, −1.73505123945761296776573171653, −0.72496175532024712643623898876,
0.72496175532024712643623898876, 1.73505123945761296776573171653, 2.66834210528100327403904088613, 3.68183590469577613779774575966, 4.64430177490141412929841993326, 5.05917114168770782977842868853, 5.87130281168689492098727740929, 6.20922550054195113709633633278, 6.77258882591034770576810068363, 7.49656327108103711519974684412
