| L(s) = 1 | + 5.39·2-s − 8.44·3-s + 21.0·4-s + 5·5-s − 45.4·6-s − 18.2·7-s + 70.3·8-s + 44.2·9-s + 26.9·10-s + 11·11-s − 177.·12-s − 77.1·13-s − 98.2·14-s − 42.2·15-s + 210.·16-s + 17.1·17-s + 238.·18-s + 19·19-s + 105.·20-s + 153.·21-s + 59.2·22-s − 197.·23-s − 594.·24-s + 25·25-s − 415.·26-s − 145.·27-s − 383.·28-s + ⋯ |
| L(s) = 1 | + 1.90·2-s − 1.62·3-s + 2.63·4-s + 0.447·5-s − 3.09·6-s − 0.984·7-s + 3.11·8-s + 1.63·9-s + 0.852·10-s + 0.301·11-s − 4.27·12-s − 1.64·13-s − 1.87·14-s − 0.726·15-s + 3.29·16-s + 0.245·17-s + 3.12·18-s + 0.229·19-s + 1.17·20-s + 1.59·21-s + 0.574·22-s − 1.79·23-s − 5.05·24-s + 0.200·25-s − 3.13·26-s − 1.03·27-s − 2.59·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 5.39T + 8T^{2} \) |
| 3 | \( 1 + 8.44T + 27T^{2} \) |
| 7 | \( 1 + 18.2T + 343T^{2} \) |
| 13 | \( 1 + 77.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 89.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 415.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 139.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 9.59T + 1.48e5T^{2} \) |
| 59 | \( 1 + 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 76.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 64.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 441.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 29.4T + 7.04e5T^{2} \) |
| 97 | \( 1 - 15.1T + 9.12e5T^{2} \) |
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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
−9.816931014056783347290285144526, −7.63952306961818028822587780683, −6.79625393197709750243709120981, −6.25914458448092005092842032970, −5.61170470183024950102902006193, −4.92510462690935198821515189111, −4.11230378820460887165149317777, −2.95380426871216263727924289277, −1.76282225887518569748456943635, 0,
1.76282225887518569748456943635, 2.95380426871216263727924289277, 4.11230378820460887165149317777, 4.92510462690935198821515189111, 5.61170470183024950102902006193, 6.25914458448092005092842032970, 6.79625393197709750243709120981, 7.63952306961818028822587780683, 9.816931014056783347290285144526
