| L(s) = 1 | − 1.26·2-s + 0.637·3-s − 0.401·4-s − 0.806·6-s + 3.31·7-s + 3.03·8-s − 2.59·9-s − 4.24·11-s − 0.255·12-s + 4.96·13-s − 4.19·14-s − 3.03·16-s − 0.765·17-s + 3.27·18-s − 0.395·19-s + 2.11·21-s + 5.36·22-s + 0.736·23-s + 1.93·24-s − 6.27·26-s − 3.56·27-s − 1.33·28-s + 0.111·29-s − 4.87·31-s − 2.23·32-s − 2.70·33-s + 0.967·34-s + ⋯ |
| L(s) = 1 | − 0.894·2-s + 0.368·3-s − 0.200·4-s − 0.329·6-s + 1.25·7-s + 1.07·8-s − 0.864·9-s − 1.27·11-s − 0.0738·12-s + 1.37·13-s − 1.12·14-s − 0.759·16-s − 0.185·17-s + 0.772·18-s − 0.0907·19-s + 0.461·21-s + 1.14·22-s + 0.153·23-s + 0.395·24-s − 1.23·26-s − 0.686·27-s − 0.251·28-s + 0.0207·29-s − 0.875·31-s − 0.394·32-s − 0.470·33-s + 0.165·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.26T + 2T^{2} \) |
| 3 | \( 1 - 0.637T + 3T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 4.96T + 13T^{2} \) |
| 17 | \( 1 + 0.765T + 17T^{2} \) |
| 19 | \( 1 + 0.395T + 19T^{2} \) |
| 23 | \( 1 - 0.736T + 23T^{2} \) |
| 29 | \( 1 - 0.111T + 29T^{2} \) |
| 31 | \( 1 + 4.87T + 31T^{2} \) |
| 37 | \( 1 - 5.92T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 0.146T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 + 4.56T + 61T^{2} \) |
| 67 | \( 1 + 9.50T + 67T^{2} \) |
| 71 | \( 1 - 1.93T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 + 3.58T + 97T^{2} \) |
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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
−7.992891610398970482842753993806, −7.52924695728461961664521116003, −6.33371082602025234996656194371, −5.48206006781278697496624933108, −4.89924285649706351060836718093, −4.07478425385127285724860563242, −3.08995602349801050683702545959, −2.08817588922685580328534059032, −1.26094946734933846211339711893, 0,
1.26094946734933846211339711893, 2.08817588922685580328534059032, 3.08995602349801050683702545959, 4.07478425385127285724860563242, 4.89924285649706351060836718093, 5.48206006781278697496624933108, 6.33371082602025234996656194371, 7.52924695728461961664521116003, 7.992891610398970482842753993806
