| L(s) = 1 | − 2-s + 2.10·3-s + 4-s + 2.46·5-s − 2.10·6-s − 3.09·7-s − 8-s + 1.41·9-s − 2.46·10-s + 2.10·12-s − 3.97·13-s + 3.09·14-s + 5.16·15-s + 16-s − 2.14·17-s − 1.41·18-s − 19-s + 2.46·20-s − 6.51·21-s − 2.55·23-s − 2.10·24-s + 1.05·25-s + 3.97·26-s − 3.33·27-s − 3.09·28-s + 6.50·29-s − 5.16·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.21·3-s + 0.5·4-s + 1.10·5-s − 0.857·6-s − 1.17·7-s − 0.353·8-s + 0.471·9-s − 0.777·10-s + 0.606·12-s − 1.10·13-s + 0.828·14-s + 1.33·15-s + 0.250·16-s − 0.520·17-s − 0.333·18-s − 0.229·19-s + 0.550·20-s − 1.42·21-s − 0.533·23-s − 0.428·24-s + 0.210·25-s + 0.780·26-s − 0.641·27-s − 0.585·28-s + 1.20·29-s − 0.943·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.10T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 23 | \( 1 + 2.55T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 - 5.48T + 41T^{2} \) |
| 43 | \( 1 - 4.59T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 6.65T + 59T^{2} \) |
| 61 | \( 1 + 6.34T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 3.17T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 + 5.47T + 97T^{2} \) |
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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
−8.046829104830908056443784397142, −7.43401045969061619519384478869, −6.47318725890374795209759494833, −6.15661283445061464128612263728, −5.03888966014034804288950689678, −3.93801106147999143223473263546, −2.82760768965359887912529260959, −2.57790312550179014743577672834, −1.62984697730654480580388959430, 0,
1.62984697730654480580388959430, 2.57790312550179014743577672834, 2.82760768965359887912529260959, 3.93801106147999143223473263546, 5.03888966014034804288950689678, 6.15661283445061464128612263728, 6.47318725890374795209759494833, 7.43401045969061619519384478869, 8.046829104830908056443784397142
