| L(s) = 1 | + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s − 12-s − 4.47·13-s + 2·15-s + 16-s + 4.47·17-s + 18-s + 6.47·19-s − 2·20-s − 23-s − 24-s − 25-s − 4.47·26-s − 27-s − 2·29-s + 2·30-s − 6.47·31-s + 32-s + 4.47·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.632·10-s − 0.288·12-s − 1.24·13-s + 0.516·15-s + 0.250·16-s + 1.08·17-s + 0.235·18-s + 1.48·19-s − 0.447·20-s − 0.208·23-s − 0.204·24-s − 0.200·25-s − 0.877·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s − 1.16·31-s + 0.176·32-s + 0.766·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 6.94T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−7.46240187672602840361923130008, −7.12384169358857544224533234657, −6.00086531050659593905619193889, −5.38527935029534650080003363422, −4.90097811643310037092845697735, −3.94143220501659481170953589650, −3.43642960235211622329161557337, −2.45568744021517076318742775169, −1.26458621099679501445844531273, 0,
1.26458621099679501445844531273, 2.45568744021517076318742775169, 3.43642960235211622329161557337, 3.94143220501659481170953589650, 4.90097811643310037092845697735, 5.38527935029534650080003363422, 6.00086531050659593905619193889, 7.12384169358857544224533234657, 7.46240187672602840361923130008
