L(s) = 1 | + 0.534·2-s − 1.71·4-s − 3.62·5-s + 4.31·7-s − 1.98·8-s − 1.93·10-s + 11-s − 1.21·13-s + 2.30·14-s + 2.36·16-s − 0.492·17-s + 0.909·19-s + 6.21·20-s + 0.534·22-s − 9.05·23-s + 8.14·25-s − 0.651·26-s − 7.39·28-s + 3.36·29-s − 1.80·31-s + 5.23·32-s − 0.263·34-s − 15.6·35-s − 2.52·37-s + 0.485·38-s + 7.19·40-s + 3.50·41-s + ⋯ |
L(s) = 1 | + 0.377·2-s − 0.857·4-s − 1.62·5-s + 1.62·7-s − 0.701·8-s − 0.612·10-s + 0.301·11-s − 0.338·13-s + 0.615·14-s + 0.592·16-s − 0.119·17-s + 0.208·19-s + 1.39·20-s + 0.113·22-s − 1.88·23-s + 1.62·25-s − 0.127·26-s − 1.39·28-s + 0.624·29-s − 0.323·31-s + 0.925·32-s − 0.0451·34-s − 2.64·35-s − 0.414·37-s + 0.0787·38-s + 1.13·40-s + 0.547·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 0.534T + 2T^{2} \) |
| 5 | \( 1 + 3.62T + 5T^{2} \) |
| 7 | \( 1 - 4.31T + 7T^{2} \) |
| 13 | \( 1 + 1.21T + 13T^{2} \) |
| 17 | \( 1 + 0.492T + 17T^{2} \) |
| 19 | \( 1 - 0.909T + 19T^{2} \) |
| 23 | \( 1 + 9.05T + 23T^{2} \) |
| 29 | \( 1 - 3.36T + 29T^{2} \) |
| 31 | \( 1 + 1.80T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 - 3.50T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 + 7.40T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 67 | \( 1 + 1.39T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 9.92T + 73T^{2} \) |
| 79 | \( 1 - 4.62T + 79T^{2} \) |
| 83 | \( 1 - 2.80T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 2.52T + 97T^{2} \) |
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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
−7.79625592107812477306183341068, −7.33417683044002266055635061071, −6.18680567632314050755750641440, −5.25872842487190860876686262902, −4.69696138871856031522078023013, −4.06738358156433451388588041710, −3.68321410838408512020511487417, −2.41689922938729550646049531448, −1.14510831013924597962284683978, 0,
1.14510831013924597962284683978, 2.41689922938729550646049531448, 3.68321410838408512020511487417, 4.06738358156433451388588041710, 4.69696138871856031522078023013, 5.25872842487190860876686262902, 6.18680567632314050755750641440, 7.33417683044002266055635061071, 7.79625592107812477306183341068