| L(s) = 1 | + 2.40·2-s − 0.476·3-s + 3.78·4-s + 5-s − 1.14·6-s + 7-s + 4.27·8-s − 2.77·9-s + 2.40·10-s + 2.16·11-s − 1.80·12-s − 4.04·13-s + 2.40·14-s − 0.476·15-s + 2.72·16-s − 5.11·17-s − 6.66·18-s − 1.75·19-s + 3.78·20-s − 0.476·21-s + 5.19·22-s − 1.21·23-s − 2.04·24-s + 25-s − 9.72·26-s + 2.75·27-s + 3.78·28-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.275·3-s + 1.89·4-s + 0.447·5-s − 0.467·6-s + 0.377·7-s + 1.51·8-s − 0.924·9-s + 0.760·10-s + 0.651·11-s − 0.520·12-s − 1.12·13-s + 0.642·14-s − 0.123·15-s + 0.682·16-s − 1.24·17-s − 1.57·18-s − 0.403·19-s + 0.845·20-s − 0.104·21-s + 1.10·22-s − 0.252·23-s − 0.416·24-s + 0.200·25-s − 1.90·26-s + 0.529·27-s + 0.714·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 2.40T + 2T^{2} \) |
| 3 | \( 1 + 0.476T + 3T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 1.75T + 19T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 + 9.33T + 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 - 4.92T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 - 0.922T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 + 1.27T + 67T^{2} \) |
| 71 | \( 1 + 6.33T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 0.363T + 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 0.346T + 97T^{2} \) |
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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
−7.06889992328054980104353269385, −6.60702533685061635732861642718, −5.84959800506399483126851631950, −5.34970493641911497846187230271, −4.76196841952618239592830734882, −4.03371573699026068883690651995, −3.29081215170177533643995124105, −2.30514431808682666434249502563, −1.89018331200514091681618177267, 0,
1.89018331200514091681618177267, 2.30514431808682666434249502563, 3.29081215170177533643995124105, 4.03371573699026068883690651995, 4.76196841952618239592830734882, 5.34970493641911497846187230271, 5.84959800506399483126851631950, 6.60702533685061635732861642718, 7.06889992328054980104353269385
