| L(s) = 1 | + 2-s − 0.104·3-s + 4-s − 0.922·5-s − 0.104·6-s + 4.49·7-s + 8-s − 2.98·9-s − 0.922·10-s − 2.62·11-s − 0.104·12-s − 2.06·13-s + 4.49·14-s + 0.0968·15-s + 16-s + 3.36·17-s − 2.98·18-s − 1.41·19-s − 0.922·20-s − 0.471·21-s − 2.62·22-s − 23-s − 0.104·24-s − 4.14·25-s − 2.06·26-s + 0.628·27-s + 4.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0605·3-s + 0.5·4-s − 0.412·5-s − 0.0428·6-s + 1.69·7-s + 0.353·8-s − 0.996·9-s − 0.291·10-s − 0.791·11-s − 0.0302·12-s − 0.573·13-s + 1.20·14-s + 0.0249·15-s + 0.250·16-s + 0.816·17-s − 0.704·18-s − 0.323·19-s − 0.206·20-s − 0.102·21-s − 0.559·22-s − 0.208·23-s − 0.0214·24-s − 0.829·25-s − 0.405·26-s + 0.120·27-s + 0.849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.104T + 3T^{2} \) |
| 5 | \( 1 + 0.922T + 5T^{2} \) |
| 7 | \( 1 - 4.49T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 + 4.17T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 + 0.924T + 41T^{2} \) |
| 43 | \( 1 - 2.50T + 43T^{2} \) |
| 47 | \( 1 + 7.26T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 1.77T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 5.27T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 0.332T + 89T^{2} \) |
| 97 | \( 1 + 7.74T + 97T^{2} \) |
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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
−7.70254646377654625003680906711, −7.23218835841673033049636258751, −5.98215857428137091672795122351, −5.40343966289960080230440822547, −4.98036014943692097781350892599, −4.13367775961983968219882173862, −3.31166405134046589117418786873, −2.34827654033084991285733660120, −1.61491699454746891231152785653, 0,
1.61491699454746891231152785653, 2.34827654033084991285733660120, 3.31166405134046589117418786873, 4.13367775961983968219882173862, 4.98036014943692097781350892599, 5.40343966289960080230440822547, 5.98215857428137091672795122351, 7.23218835841673033049636258751, 7.70254646377654625003680906711
