| L(s) = 1 | + 0.300·2-s + 2.68·3-s − 1.90·4-s + 5-s + 0.806·6-s − 7-s − 1.17·8-s + 4.18·9-s + 0.300·10-s − 5.11·12-s − 3.71·13-s − 0.300·14-s + 2.68·15-s + 3.46·16-s − 2.57·17-s + 1.25·18-s − 4.28·19-s − 1.90·20-s − 2.68·21-s − 6.57·23-s − 3.15·24-s + 25-s − 1.11·26-s + 3.18·27-s + 1.90·28-s + 5.39·29-s + 0.806·30-s + ⋯ |
| L(s) = 1 | + 0.212·2-s + 1.54·3-s − 0.954·4-s + 0.447·5-s + 0.329·6-s − 0.377·7-s − 0.415·8-s + 1.39·9-s + 0.0950·10-s − 1.47·12-s − 1.02·13-s − 0.0803·14-s + 0.692·15-s + 0.866·16-s − 0.625·17-s + 0.296·18-s − 0.982·19-s − 0.426·20-s − 0.585·21-s − 1.37·23-s − 0.643·24-s + 0.200·25-s − 0.218·26-s + 0.613·27-s + 0.360·28-s + 1.00·29-s + 0.147·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.300T + 2T^{2} \) |
| 3 | \( 1 - 2.68T + 3T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + 4.28T + 19T^{2} \) |
| 23 | \( 1 + 6.57T + 23T^{2} \) |
| 29 | \( 1 - 5.39T + 29T^{2} \) |
| 31 | \( 1 - 0.259T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 + 5.56T + 41T^{2} \) |
| 43 | \( 1 + 7.87T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 9.55T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 5.34T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 6.53T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−8.368391319320508530450310854454, −7.51661284601716791132604677636, −6.59993424469940915286358978591, −5.82791426288517442065363808662, −4.66580842458492710850646324819, −4.27727981340776338807403344506, −3.28912764668347652667513150028, −2.61400111471674258097005765193, −1.75021700119151168519816621475, 0,
1.75021700119151168519816621475, 2.61400111471674258097005765193, 3.28912764668347652667513150028, 4.27727981340776338807403344506, 4.66580842458492710850646324819, 5.82791426288517442065363808662, 6.59993424469940915286358978591, 7.51661284601716791132604677636, 8.368391319320508530450310854454
