| L(s) = 1 | − 2-s + 0.503·3-s + 4-s + 1.68·5-s − 0.503·6-s + 4.13·7-s − 8-s − 2.74·9-s − 1.68·10-s + 0.105·11-s + 0.503·12-s − 4.13·14-s + 0.848·15-s + 16-s − 7.67·17-s + 2.74·18-s + 19-s + 1.68·20-s + 2.08·21-s − 0.105·22-s + 2.21·23-s − 0.503·24-s − 2.15·25-s − 2.89·27-s + 4.13·28-s + 2.83·29-s − 0.848·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.290·3-s + 0.5·4-s + 0.753·5-s − 0.205·6-s + 1.56·7-s − 0.353·8-s − 0.915·9-s − 0.532·10-s + 0.0317·11-s + 0.145·12-s − 1.10·14-s + 0.219·15-s + 0.250·16-s − 1.86·17-s + 0.647·18-s + 0.229·19-s + 0.376·20-s + 0.454·21-s − 0.0224·22-s + 0.462·23-s − 0.102·24-s − 0.431·25-s − 0.556·27-s + 0.782·28-s + 0.526·29-s − 0.154·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.503T + 3T^{2} \) |
| 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 - 0.105T + 11T^{2} \) |
| 17 | \( 1 + 7.67T + 17T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 8.42T + 37T^{2} \) |
| 41 | \( 1 - 1.72T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 - 8.40T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 9.69T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.945075807628931136804128573289, −7.01152311472892215585979876321, −6.42978398788149008787847528856, −5.42522744795762582346024673770, −5.04279829245712002014408013974, −3.99838776426179888769108610707, −2.87701359657148046035169844360, −2.01934240350280220250588831749, −1.58326577410376052645930227966, 0,
1.58326577410376052645930227966, 2.01934240350280220250588831749, 2.87701359657148046035169844360, 3.99838776426179888769108610707, 5.04279829245712002014408013974, 5.42522744795762582346024673770, 6.42978398788149008787847528856, 7.01152311472892215585979876321, 7.945075807628931136804128573289
