| L(s) = 1 | + 2.09·2-s − 3-s + 2.40·4-s − 3.89·5-s − 2.09·6-s + 7-s + 0.859·8-s + 9-s − 8.18·10-s − 2.92·11-s − 2.40·12-s + 3.41·13-s + 2.09·14-s + 3.89·15-s − 3.01·16-s − 6.66·17-s + 2.09·18-s − 6.95·19-s − 9.38·20-s − 21-s − 6.14·22-s − 6.75·23-s − 0.859·24-s + 10.1·25-s + 7.17·26-s − 27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 0.577·3-s + 1.20·4-s − 1.74·5-s − 0.857·6-s + 0.377·7-s + 0.303·8-s + 0.333·9-s − 2.58·10-s − 0.882·11-s − 0.695·12-s + 0.947·13-s + 0.561·14-s + 1.00·15-s − 0.753·16-s − 1.61·17-s + 0.494·18-s − 1.59·19-s − 2.09·20-s − 0.218·21-s − 1.31·22-s − 1.40·23-s − 0.175·24-s + 2.03·25-s + 1.40·26-s − 0.192·27-s + 0.455·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 2.09T + 2T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 6.66T + 17T^{2} \) |
| 19 | \( 1 + 6.95T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 8.13T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 - 4.72T + 53T^{2} \) |
| 59 | \( 1 - 6.57T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 - 3.11T + 71T^{2} \) |
| 73 | \( 1 + 3.37T + 73T^{2} \) |
| 79 | \( 1 - 0.388T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 - 1.50T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−10.51440656506158926285558488063, −8.591074500573775882948810465449, −8.170137377123793391209235949722, −6.88263599479949070822142644667, −6.29835145018245947465178040100, −5.07653920475216648487411433897, −4.29211710614106725633861121696, −3.86588035761105296039027722876, −2.44898604832710716284751517981, 0,
2.44898604832710716284751517981, 3.86588035761105296039027722876, 4.29211710614106725633861121696, 5.07653920475216648487411433897, 6.29835145018245947465178040100, 6.88263599479949070822142644667, 8.170137377123793391209235949722, 8.591074500573775882948810465449, 10.51440656506158926285558488063
