| L(s) = 1 | + 1.13·2-s + 2.48·3-s − 0.713·4-s + 5-s + 2.81·6-s − 3.07·8-s + 3.18·9-s + 1.13·10-s − 3.49·11-s − 1.77·12-s − 0.596·13-s + 2.48·15-s − 2.06·16-s + 4.26·17-s + 3.60·18-s − 2.77·19-s − 0.713·20-s − 3.96·22-s + 4.86·23-s − 7.65·24-s + 25-s − 0.676·26-s + 0.450·27-s + 4.26·29-s + 2.81·30-s + 31-s + 3.81·32-s + ⋯ |
| L(s) = 1 | + 0.801·2-s + 1.43·3-s − 0.356·4-s + 0.447·5-s + 1.15·6-s − 1.08·8-s + 1.06·9-s + 0.358·10-s − 1.05·11-s − 0.512·12-s − 0.165·13-s + 0.641·15-s − 0.515·16-s + 1.03·17-s + 0.850·18-s − 0.637·19-s − 0.159·20-s − 0.844·22-s + 1.01·23-s − 1.56·24-s + 0.200·25-s − 0.132·26-s + 0.0867·27-s + 0.792·29-s + 0.514·30-s + 0.179·31-s + 0.674·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.613735136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.613735136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.596T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 4.86T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 - 6.20T + 41T^{2} \) |
| 43 | \( 1 - 9.56T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 7.47T + 53T^{2} \) |
| 59 | \( 1 + 0.643T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 - 3.34T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 + 3.37T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 - 2.40T + 89T^{2} \) |
| 97 | \( 1 - 0.815T + 97T^{2} \) |
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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
−7.899111775894256752521617955308, −7.39776498409009047188879909689, −6.34055496815967153571803418560, −5.63167493468257484713317448501, −4.96681347638518939057842128917, −4.23209271714560517061174952024, −3.46726589248076717714635366510, −2.72598942630976226596715624251, −2.36603657397934869638056887969, −0.888991129363155098323382315536,
0.888991129363155098323382315536, 2.36603657397934869638056887969, 2.72598942630976226596715624251, 3.46726589248076717714635366510, 4.23209271714560517061174952024, 4.96681347638518939057842128917, 5.63167493468257484713317448501, 6.34055496815967153571803418560, 7.39776498409009047188879909689, 7.899111775894256752521617955308
