| L(s) = 1 | − 5·5-s − 13.7·7-s − 21.6·11-s − 79.6·13-s + 64.3·17-s − 144.·19-s + 26.9·23-s + 25·25-s − 297.·29-s − 96.6·31-s + 68.6·35-s + 401.·37-s + 11.6·41-s + 292.·43-s − 175.·47-s − 154.·49-s − 155.·53-s + 108.·55-s − 595.·59-s − 62.1·61-s + 398.·65-s − 160.·67-s + 959.·71-s + 763.·73-s + 297.·77-s + 63.9·79-s − 917.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.741·7-s − 0.593·11-s − 1.69·13-s + 0.918·17-s − 1.74·19-s + 0.244·23-s + 0.200·25-s − 1.90·29-s − 0.559·31-s + 0.331·35-s + 1.78·37-s + 0.0445·41-s + 1.03·43-s − 0.544·47-s − 0.450·49-s − 0.402·53-s + 0.265·55-s − 1.31·59-s − 0.130·61-s + 0.759·65-s − 0.292·67-s + 1.60·71-s + 1.22·73-s + 0.439·77-s + 0.0910·79-s − 1.21·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6457916172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6457916172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 + 13.7T + 343T^{2} \) |
| 11 | \( 1 + 21.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 79.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 297.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 96.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 401.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 11.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 292.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 175.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 595.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 62.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 160.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 959.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 63.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 917.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 128.T + 9.12e5T^{2} \) |
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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
−9.273559638279700470607475550344, −7.977049687378150231834805646281, −7.59200589556325111531705795736, −6.67412740598814228921078783576, −5.77491269873650536578476233678, −4.86877840144332213006173628090, −3.97507822096900136832005260907, −2.95430044150271826404397509887, −2.07951230231892802828974818518, −0.36181427718427167762982813980,
0.36181427718427167762982813980, 2.07951230231892802828974818518, 2.95430044150271826404397509887, 3.97507822096900136832005260907, 4.86877840144332213006173628090, 5.77491269873650536578476233678, 6.67412740598814228921078783576, 7.59200589556325111531705795736, 7.977049687378150231834805646281, 9.273559638279700470607475550344
