| L(s) = 1 | + 1.44·2-s − 1.55·3-s + 0.0964·4-s − 5-s − 2.24·6-s − 7-s − 2.75·8-s − 0.595·9-s − 1.44·10-s − 2.88·11-s − 0.149·12-s + 5.15·13-s − 1.44·14-s + 1.55·15-s − 4.18·16-s + 1.97·17-s − 0.861·18-s + 0.738·19-s − 0.0964·20-s + 1.55·21-s − 4.17·22-s + 5.34·23-s + 4.27·24-s + 25-s + 7.46·26-s + 5.57·27-s − 0.0964·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s − 0.895·3-s + 0.0482·4-s − 0.447·5-s − 0.916·6-s − 0.377·7-s − 0.974·8-s − 0.198·9-s − 0.457·10-s − 0.868·11-s − 0.0431·12-s + 1.42·13-s − 0.386·14-s + 0.400·15-s − 1.04·16-s + 0.478·17-s − 0.203·18-s + 0.169·19-s − 0.0215·20-s + 0.338·21-s − 0.889·22-s + 1.11·23-s + 0.872·24-s + 0.200·25-s + 1.46·26-s + 1.07·27-s − 0.0182·28-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 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 - 0.738T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 + 5.24T + 29T^{2} \) |
| 31 | \( 1 + 0.745T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 - 1.08T + 43T^{2} \) |
| 47 | \( 1 - 7.63T + 47T^{2} \) |
| 53 | \( 1 - 0.105T + 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 0.471T + 67T^{2} \) |
| 71 | \( 1 + 0.183T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 2.82T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 6.81T + 97T^{2} \) |
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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
−7.32301952133088532439785354533, −6.44506298303533586936065125779, −5.90652859600726421832476335849, −5.39597049685533135754112818721, −4.82875731088485721039504866364, −3.91734618066792112667696662724, −3.34683533133933400382473982236, −2.62027814273954896297807619429, −1.02736789865153483315531878250, 0,
1.02736789865153483315531878250, 2.62027814273954896297807619429, 3.34683533133933400382473982236, 3.91734618066792112667696662724, 4.82875731088485721039504866364, 5.39597049685533135754112818721, 5.90652859600726421832476335849, 6.44506298303533586936065125779, 7.32301952133088532439785354533
