| L(s) = 1 | + 0.0253·2-s − 3·3-s − 7.99·4-s + 8.85·5-s − 0.0759·6-s + 13.8·7-s − 0.405·8-s + 9·9-s + 0.224·10-s − 33.5·11-s + 23.9·12-s − 15.9·13-s + 0.351·14-s − 26.5·15-s + 63.9·16-s − 14.1·17-s + 0.227·18-s − 111.·19-s − 70.8·20-s − 41.6·21-s − 0.850·22-s − 188.·23-s + 1.21·24-s − 46.5·25-s − 0.403·26-s − 27·27-s − 110.·28-s + ⋯ |
| L(s) = 1 | + 0.00895·2-s − 0.577·3-s − 0.999·4-s + 0.792·5-s − 0.00516·6-s + 0.749·7-s − 0.0179·8-s + 0.333·9-s + 0.00709·10-s − 0.920·11-s + 0.577·12-s − 0.339·13-s + 0.00670·14-s − 0.457·15-s + 0.999·16-s − 0.202·17-s + 0.00298·18-s − 1.35·19-s − 0.791·20-s − 0.432·21-s − 0.00824·22-s − 1.71·23-s + 0.0103·24-s − 0.372·25-s − 0.00304·26-s − 0.192·27-s − 0.749·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 0.0253T + 8T^{2} \) |
| 5 | \( 1 - 8.85T + 125T^{2} \) |
| 7 | \( 1 - 13.8T + 343T^{2} \) |
| 11 | \( 1 + 33.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 56.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 53.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 344.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 291.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 636.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 489.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 91.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 179.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 414.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 889.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 812.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 623.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.22e3T + 9.12e5T^{2} \) |
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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
−11.80838481604954969183156046538, −10.50544302499193807901332203984, −9.936836182178608323172351401943, −8.682757238213593302717555100404, −7.73871437727288268444485329254, −6.08959625881058652956953784163, −5.21201927568492211947405715683, −4.19778645602240867787858310538, −1.99700688626211368875194327715, 0,
1.99700688626211368875194327715, 4.19778645602240867787858310538, 5.21201927568492211947405715683, 6.08959625881058652956953784163, 7.73871437727288268444485329254, 8.682757238213593302717555100404, 9.936836182178608323172351401943, 10.50544302499193807901332203984, 11.80838481604954969183156046538
