| L(s) = 1 | − 1.77·2-s − 3-s + 1.13·4-s − 2.58·5-s + 1.77·6-s + 2.67·7-s + 1.52·8-s + 9-s + 4.57·10-s − 2.53·11-s − 1.13·12-s + 13-s − 4.74·14-s + 2.58·15-s − 4.98·16-s + 2.43·17-s − 1.77·18-s − 1.52·19-s − 2.94·20-s − 2.67·21-s + 4.48·22-s − 5.39·23-s − 1.52·24-s + 1.65·25-s − 1.77·26-s − 27-s + 3.05·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.569·4-s − 1.15·5-s + 0.723·6-s + 1.01·7-s + 0.538·8-s + 0.333·9-s + 1.44·10-s − 0.763·11-s − 0.328·12-s + 0.277·13-s − 1.26·14-s + 0.666·15-s − 1.24·16-s + 0.590·17-s − 0.417·18-s − 0.349·19-s − 0.657·20-s − 0.584·21-s + 0.956·22-s − 1.12·23-s − 0.311·24-s + 0.331·25-s − 0.347·26-s − 0.192·27-s + 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 + 2.53T + 11T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 - 9.71T + 29T^{2} \) |
| 31 | \( 1 - 0.543T + 31T^{2} \) |
| 37 | \( 1 - 9.45T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 8.84T + 47T^{2} \) |
| 53 | \( 1 - 0.414T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 + 9.29T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 0.333T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.115595675187279961469825631472, −7.78735210476888322688533746638, −6.88858544543732502307458666114, −6.02072818426361399576856111681, −4.78763456660166018360819091264, −4.57641494070721400036562574848, −3.41368981037844910611216775963, −2.05736840940957626942040492879, −1.02494804147695783788789292575, 0,
1.02494804147695783788789292575, 2.05736840940957626942040492879, 3.41368981037844910611216775963, 4.57641494070721400036562574848, 4.78763456660166018360819091264, 6.02072818426361399576856111681, 6.88858544543732502307458666114, 7.78735210476888322688533746638, 8.115595675187279961469825631472
