| L(s) = 1 | − 1.71·2-s − 3-s + 0.956·4-s + 1.71·6-s + 0.918·7-s + 1.79·8-s + 9-s + 6.16·11-s − 0.956·12-s − 2.75·13-s − 1.57·14-s − 4.99·16-s − 0.211·17-s − 1.71·18-s + 2.34·19-s − 0.918·21-s − 10.5·22-s + 2.13·23-s − 1.79·24-s + 4.72·26-s − 27-s + 0.878·28-s − 5.05·29-s + 1.03·31-s + 5.00·32-s − 6.16·33-s + 0.364·34-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.478·4-s + 0.701·6-s + 0.347·7-s + 0.634·8-s + 0.333·9-s + 1.85·11-s − 0.275·12-s − 0.762·13-s − 0.422·14-s − 1.24·16-s − 0.0513·17-s − 0.405·18-s + 0.538·19-s − 0.200·21-s − 2.25·22-s + 0.444·23-s − 0.366·24-s + 0.927·26-s − 0.192·27-s + 0.166·28-s − 0.937·29-s + 0.185·31-s + 0.884·32-s − 1.07·33-s + 0.0624·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 67 | \( 1 + T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 7 | \( 1 - 0.918T + 7T^{2} \) |
| 11 | \( 1 - 6.16T + 11T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 + 0.211T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 + 6.05T + 37T^{2} \) |
| 41 | \( 1 - 0.346T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 1.83T + 47T^{2} \) |
| 53 | \( 1 + 8.65T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 71 | \( 1 + 7.36T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 8.97T + 79T^{2} \) |
| 83 | \( 1 + 0.201T + 83T^{2} \) |
| 89 | \( 1 + 8.33T + 89T^{2} \) |
| 97 | \( 1 - 4.46T + 97T^{2} \) |
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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
−7.910476073186289933160242078194, −7.22127077135389933985044916591, −6.72338794915698681309066108372, −5.86133590962864129281514912292, −4.85539916938516955336918623304, −4.31683016842500622229716290543, −3.26845235851591102061322771346, −1.79391246949898079267577709353, −1.24732157620101089343286950574, 0,
1.24732157620101089343286950574, 1.79391246949898079267577709353, 3.26845235851591102061322771346, 4.31683016842500622229716290543, 4.85539916938516955336918623304, 5.86133590962864129281514912292, 6.72338794915698681309066108372, 7.22127077135389933985044916591, 7.910476073186289933160242078194
