| L(s) = 1 | + 2-s − 3-s + 4-s − 2.98·5-s − 6-s + 3.36·7-s + 8-s + 9-s − 2.98·10-s + 3.96·11-s − 12-s − 13-s + 3.36·14-s + 2.98·15-s + 16-s + 1.88·17-s + 18-s − 7.32·19-s − 2.98·20-s − 3.36·21-s + 3.96·22-s − 3.99·23-s − 24-s + 3.93·25-s − 26-s − 27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.33·5-s − 0.408·6-s + 1.27·7-s + 0.353·8-s + 0.333·9-s − 0.945·10-s + 1.19·11-s − 0.288·12-s − 0.277·13-s + 0.900·14-s + 0.772·15-s + 0.250·16-s + 0.457·17-s + 0.235·18-s − 1.67·19-s − 0.668·20-s − 0.734·21-s + 0.844·22-s − 0.833·23-s − 0.204·24-s + 0.787·25-s − 0.196·26-s − 0.192·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 2.98T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + 7.32T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 8.71T + 29T^{2} \) |
| 31 | \( 1 - 1.96T + 31T^{2} \) |
| 37 | \( 1 + 2.32T + 37T^{2} \) |
| 41 | \( 1 - 6.61T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 8.39T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 + 0.371T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 3.43T + 67T^{2} \) |
| 71 | \( 1 - 3.20T + 71T^{2} \) |
| 73 | \( 1 - 0.938T + 73T^{2} \) |
| 79 | \( 1 + 4.29T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 1.18T + 97T^{2} \) |
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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
−7.59823116431643798532353776214, −6.70077499342001540009394623038, −6.08184696707471867771689757726, −5.27628501885124373220090924857, −4.38801097113779816043964855183, −4.20764902491245729679678046478, −3.47928751084123299151399708836, −2.14470652192325889387839791027, −1.35934981427904192063501542386, 0,
1.35934981427904192063501542386, 2.14470652192325889387839791027, 3.47928751084123299151399708836, 4.20764902491245729679678046478, 4.38801097113779816043964855183, 5.27628501885124373220090924857, 6.08184696707471867771689757726, 6.70077499342001540009394623038, 7.59823116431643798532353776214
