| L(s) = 1 | − 0.602·2-s − 3.43·3-s − 1.63·4-s − 5-s + 2.07·6-s − 0.349·7-s + 2.19·8-s + 8.83·9-s + 0.602·10-s + 11-s + 5.63·12-s − 6.76·13-s + 0.210·14-s + 3.43·15-s + 1.95·16-s − 3.51·17-s − 5.32·18-s − 2.53·19-s + 1.63·20-s + 1.20·21-s − 0.602·22-s − 1.89·23-s − 7.53·24-s + 25-s + 4.07·26-s − 20.0·27-s + 0.571·28-s + ⋯ |
| L(s) = 1 | − 0.426·2-s − 1.98·3-s − 0.818·4-s − 0.447·5-s + 0.846·6-s − 0.132·7-s + 0.774·8-s + 2.94·9-s + 0.190·10-s + 0.301·11-s + 1.62·12-s − 1.87·13-s + 0.0562·14-s + 0.888·15-s + 0.488·16-s − 0.852·17-s − 1.25·18-s − 0.582·19-s + 0.366·20-s + 0.262·21-s − 0.128·22-s − 0.395·23-s − 1.53·24-s + 0.200·25-s + 0.799·26-s − 3.86·27-s + 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 0.602T + 2T^{2} \) |
| 3 | \( 1 + 3.43T + 3T^{2} \) |
| 7 | \( 1 + 0.349T + 7T^{2} \) |
| 13 | \( 1 + 6.76T + 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 9.67T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 7.66T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 - 9.34T + 43T^{2} \) |
| 47 | \( 1 - 0.371T + 47T^{2} \) |
| 53 | \( 1 - 5.85T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 + 2.51T + 61T^{2} \) |
| 67 | \( 1 - 8.73T + 67T^{2} \) |
| 71 | \( 1 + 4.53T + 71T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 3.71T + 89T^{2} \) |
| 97 | \( 1 + 3.94T + 97T^{2} \) |
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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
−7.951106445046879322144306599985, −7.14129840405961346623714948965, −6.74231394333972363048016097988, −5.76088275230072773923469678327, −4.97129664710007605328573222271, −4.56579908198877134445631880904, −3.88837342644812398567502168834, −2.11755976889190947265891363968, −0.77964629564444377846352435547, 0,
0.77964629564444377846352435547, 2.11755976889190947265891363968, 3.88837342644812398567502168834, 4.56579908198877134445631880904, 4.97129664710007605328573222271, 5.76088275230072773923469678327, 6.74231394333972363048016097988, 7.14129840405961346623714948965, 7.951106445046879322144306599985
