| L(s) = 1 | + 1.31·2-s − 0.792·3-s − 0.268·4-s + 5-s − 1.04·6-s − 1.49·7-s − 2.98·8-s − 2.37·9-s + 1.31·10-s + 4.87·11-s + 0.213·12-s + 5.90·13-s − 1.97·14-s − 0.792·15-s − 3.38·16-s + 0.168·17-s − 3.12·18-s − 0.539·19-s − 0.268·20-s + 1.18·21-s + 6.40·22-s + 1.93·23-s + 2.36·24-s + 25-s + 7.77·26-s + 4.25·27-s + 0.402·28-s + ⋯ |
| L(s) = 1 | + 0.930·2-s − 0.457·3-s − 0.134·4-s + 0.447·5-s − 0.425·6-s − 0.566·7-s − 1.05·8-s − 0.790·9-s + 0.416·10-s + 1.46·11-s + 0.0615·12-s + 1.63·13-s − 0.526·14-s − 0.204·15-s − 0.847·16-s + 0.0407·17-s − 0.735·18-s − 0.123·19-s − 0.0601·20-s + 0.259·21-s + 1.36·22-s + 0.403·23-s + 0.482·24-s + 0.200·25-s + 1.52·26-s + 0.819·27-s + 0.0761·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.074085521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.074085521\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 3 | \( 1 + 0.792T + 3T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 4.87T + 11T^{2} \) |
| 13 | \( 1 - 5.90T + 13T^{2} \) |
| 17 | \( 1 - 0.168T + 17T^{2} \) |
| 19 | \( 1 + 0.539T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 + 2.81T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.486T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 + 4.21T + 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 + 8.47T + 97T^{2} \) |
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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
−9.537822423794000410734875797827, −9.040685549322164384879177831668, −8.330118780131774674114635005566, −6.68805630357928584132809829555, −6.17568053894684408450304260077, −5.69605593960259062611312488486, −4.53068044584795278122830978253, −3.71026349108325531446030801824, −2.84013986539803896652886752079, −1.01031541735464911419795025138,
1.01031541735464911419795025138, 2.84013986539803896652886752079, 3.71026349108325531446030801824, 4.53068044584795278122830978253, 5.69605593960259062611312488486, 6.17568053894684408450304260077, 6.68805630357928584132809829555, 8.330118780131774674114635005566, 9.040685549322164384879177831668, 9.537822423794000410734875797827
