| L(s) = 1 | − 2-s + 2.16·3-s + 4-s + 2.31·5-s − 2.16·6-s + 7-s − 8-s + 1.68·9-s − 2.31·10-s − 2.02·11-s + 2.16·12-s − 6.49·13-s − 14-s + 5.01·15-s + 16-s − 0.702·17-s − 1.68·18-s + 5.38·19-s + 2.31·20-s + 2.16·21-s + 2.02·22-s − 5.42·23-s − 2.16·24-s + 0.373·25-s + 6.49·26-s − 2.84·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.24·3-s + 0.5·4-s + 1.03·5-s − 0.883·6-s + 0.377·7-s − 0.353·8-s + 0.562·9-s − 0.733·10-s − 0.611·11-s + 0.624·12-s − 1.80·13-s − 0.267·14-s + 1.29·15-s + 0.250·16-s − 0.170·17-s − 0.397·18-s + 1.23·19-s + 0.518·20-s + 0.472·21-s + 0.432·22-s − 1.13·23-s − 0.441·24-s + 0.0747·25-s + 1.27·26-s − 0.547·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 - 2.16T + 3T^{2} \) |
| 5 | \( 1 - 2.31T + 5T^{2} \) |
| 11 | \( 1 + 2.02T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 + 0.702T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.20T + 43T^{2} \) |
| 47 | \( 1 + 2.99T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 + 4.54T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + 5.32T + 89T^{2} \) |
| 97 | \( 1 - 0.483T + 97T^{2} \) |
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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
−7.82428013483935739930229779303, −7.35031550795373401707752950454, −6.54251199484783433181088551929, −5.38030310497592392149285568259, −5.13739641763408985205356835003, −3.73087774835695854284037910071, −2.93219964423426593500201827310, −2.08499813131106352798642335633, −1.81992414827105645156582213637, 0,
1.81992414827105645156582213637, 2.08499813131106352798642335633, 2.93219964423426593500201827310, 3.73087774835695854284037910071, 5.13739641763408985205356835003, 5.38030310497592392149285568259, 6.54251199484783433181088551929, 7.35031550795373401707752950454, 7.82428013483935739930229779303
