L(s) = 1 | − 2-s − 0.785·3-s + 4-s − 1.42·5-s + 0.785·6-s + 3.87·7-s − 8-s − 2.38·9-s + 1.42·10-s + 4.24·11-s − 0.785·12-s + 2.46·13-s − 3.87·14-s + 1.11·15-s + 16-s + 1.55·17-s + 2.38·18-s − 2.53·19-s − 1.42·20-s − 3.04·21-s − 4.24·22-s − 23-s + 0.785·24-s − 2.97·25-s − 2.46·26-s + 4.22·27-s + 3.87·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.453·3-s + 0.5·4-s − 0.636·5-s + 0.320·6-s + 1.46·7-s − 0.353·8-s − 0.794·9-s + 0.449·10-s + 1.28·11-s − 0.226·12-s + 0.684·13-s − 1.03·14-s + 0.288·15-s + 0.250·16-s + 0.376·17-s + 0.561·18-s − 0.581·19-s − 0.318·20-s − 0.664·21-s − 0.905·22-s − 0.208·23-s + 0.160·24-s − 0.595·25-s − 0.483·26-s + 0.813·27-s + 0.732·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 0.785T + 3T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 + 9.30T + 43T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 - 9.88T + 67T^{2} \) |
| 71 | \( 1 - 6.27T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 8.91T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.025789971808667408500548922619, −7.02514469904903957391448443901, −6.45755012974403635298279615357, −5.61657805371974909869042490042, −4.92069859225941676131467131883, −3.99643880645125027113445767435, −3.30376462166829058128289419170, −1.92061952969320063084201513158, −1.28773358632734674168327701468, 0,
1.28773358632734674168327701468, 1.92061952969320063084201513158, 3.30376462166829058128289419170, 3.99643880645125027113445767435, 4.92069859225941676131467131883, 5.61657805371974909869042490042, 6.45755012974403635298279615357, 7.02514469904903957391448443901, 8.025789971808667408500548922619