L(s) = 1 | + 0.364·2-s − 1.86·4-s + 2.96·5-s + 5.11·7-s − 1.41·8-s + 1.08·10-s − 0.626·11-s + 6.35·13-s + 1.86·14-s + 3.21·16-s + 2.99·17-s − 1.47·19-s − 5.52·20-s − 0.228·22-s − 5.28·23-s + 3.76·25-s + 2.31·26-s − 9.54·28-s + 3.00·29-s + 0.625·31-s + 3.99·32-s + 1.09·34-s + 15.1·35-s + 3.27·37-s − 0.538·38-s − 4.17·40-s − 9.49·41-s + ⋯ |
L(s) = 1 | + 0.258·2-s − 0.933·4-s + 1.32·5-s + 1.93·7-s − 0.498·8-s + 0.341·10-s − 0.188·11-s + 1.76·13-s + 0.499·14-s + 0.804·16-s + 0.725·17-s − 0.338·19-s − 1.23·20-s − 0.0487·22-s − 1.10·23-s + 0.752·25-s + 0.454·26-s − 1.80·28-s + 0.558·29-s + 0.112·31-s + 0.706·32-s + 0.187·34-s + 2.55·35-s + 0.538·37-s − 0.0873·38-s − 0.660·40-s − 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.792948812\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.792948812\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
good | 2 | \( 1 - 0.364T + 2T^{2} \) |
| 5 | \( 1 - 2.96T + 5T^{2} \) |
| 7 | \( 1 - 5.11T + 7T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 - 6.35T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 5.28T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 - 0.625T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 + 9.49T + 41T^{2} \) |
| 43 | \( 1 + 7.31T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 0.986T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 9.13T + 67T^{2} \) |
| 71 | \( 1 + 8.78T + 71T^{2} \) |
| 73 | \( 1 + 0.0828T + 73T^{2} \) |
| 79 | \( 1 + 9.61T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−8.852282690982646522657028632291, −8.406303455134486436451901310577, −7.88106217102542737326355959566, −6.41476372443912363692871220009, −5.73583916853559048134079115876, −5.12897403118756489863939620726, −4.39562321921873373847570774338, −3.39591597117171296948026951794, −1.91529930560960754256969032531, −1.24272980920415903753695849247,
1.24272980920415903753695849247, 1.91529930560960754256969032531, 3.39591597117171296948026951794, 4.39562321921873373847570774338, 5.12897403118756489863939620726, 5.73583916853559048134079115876, 6.41476372443912363692871220009, 7.88106217102542737326355959566, 8.406303455134486436451901310577, 8.852282690982646522657028632291