L(s) = 1 | − 2-s − 2.24·3-s + 4-s + 5-s + 2.24·6-s + 7-s − 8-s + 2.05·9-s − 10-s − 2.24·12-s − 0.941·13-s − 14-s − 2.24·15-s + 16-s − 6.49·17-s − 2.05·18-s + 4.36·19-s + 20-s − 2.24·21-s + 6.24·23-s + 2.24·24-s + 25-s + 0.941·26-s + 2.11·27-s + 28-s − 8.74·29-s + 2.24·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.447·5-s + 0.918·6-s + 0.377·7-s − 0.353·8-s + 0.686·9-s − 0.316·10-s − 0.649·12-s − 0.261·13-s − 0.267·14-s − 0.580·15-s + 0.250·16-s − 1.57·17-s − 0.485·18-s + 1.00·19-s + 0.223·20-s − 0.490·21-s + 1.30·23-s + 0.459·24-s + 0.200·25-s + 0.184·26-s + 0.407·27-s + 0.188·28-s − 1.62·29-s + 0.410·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6923054243\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6923054243\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.24T + 3T^{2} \) |
| 13 | \( 1 + 0.941T + 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 + 9.55T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 - 2.13T + 41T^{2} \) |
| 43 | \( 1 + 7.67T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 4.74T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 - 8.49T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.48695660552970232118041461455, −7.19558674327580512944838775717, −6.40966989065172094313639560424, −5.74906769180087884411473950460, −5.19462346015292754778413417257, −4.55204984954478532663162877535, −3.42654582609784171412374442967, −2.34779885784852404045892139007, −1.53216886223004944863851438552, −0.49752383748670777740194383243,
0.49752383748670777740194383243, 1.53216886223004944863851438552, 2.34779885784852404045892139007, 3.42654582609784171412374442967, 4.55204984954478532663162877535, 5.19462346015292754778413417257, 5.74906769180087884411473950460, 6.40966989065172094313639560424, 7.19558674327580512944838775717, 7.48695660552970232118041461455