| L(s) = 1 | − 1.92·2-s + 0.557·3-s + 1.69·4-s − 1.07·6-s − 0.976·7-s + 0.583·8-s − 2.68·9-s + 5.68·11-s + 0.945·12-s − 0.519·13-s + 1.87·14-s − 4.51·16-s + 2.78·17-s + 5.17·18-s − 2.82·19-s − 0.544·21-s − 10.9·22-s + 7.25·23-s + 0.325·24-s + 0.997·26-s − 3.17·27-s − 1.65·28-s − 2.98·29-s − 4.59·31-s + 7.51·32-s + 3.16·33-s − 5.36·34-s + ⋯ |
| L(s) = 1 | − 1.35·2-s + 0.321·3-s + 0.848·4-s − 0.437·6-s − 0.369·7-s + 0.206·8-s − 0.896·9-s + 1.71·11-s + 0.273·12-s − 0.143·13-s + 0.501·14-s − 1.12·16-s + 0.676·17-s + 1.21·18-s − 0.647·19-s − 0.118·21-s − 2.32·22-s + 1.51·23-s + 0.0664·24-s + 0.195·26-s − 0.610·27-s − 0.313·28-s − 0.554·29-s − 0.825·31-s + 1.32·32-s + 0.551·33-s − 0.919·34-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 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 - 0.557T + 3T^{2} \) |
| 7 | \( 1 + 0.976T + 7T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 + 0.519T + 13T^{2} \) |
| 17 | \( 1 - 2.78T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 + 2.98T + 29T^{2} \) |
| 31 | \( 1 + 4.59T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 1.99T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 - 1.33T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 1.00T + 97T^{2} \) |
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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
−7.913340830435457690874929074632, −7.09503543410657988520707081147, −6.67126856317789474578383607429, −5.77767507891669814762596703855, −4.85107250723875226765995610882, −3.79820422053011898737099971829, −3.14593654742231427822651706252, −1.99555794776677650083119770137, −1.18850553453389439771854880373, 0,
1.18850553453389439771854880373, 1.99555794776677650083119770137, 3.14593654742231427822651706252, 3.79820422053011898737099971829, 4.85107250723875226765995610882, 5.77767507891669814762596703855, 6.67126856317789474578383607429, 7.09503543410657988520707081147, 7.913340830435457690874929074632
