| L(s) = 1 | + 2-s + 2.25·3-s + 4-s + 5-s + 2.25·6-s + 5.00·7-s + 8-s + 2.07·9-s + 10-s + 2.54·11-s + 2.25·12-s − 0.683·13-s + 5.00·14-s + 2.25·15-s + 16-s − 3.71·17-s + 2.07·18-s + 1.37·19-s + 20-s + 11.2·21-s + 2.54·22-s − 1.36·23-s + 2.25·24-s + 25-s − 0.683·26-s − 2.07·27-s + 5.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.447·5-s + 0.920·6-s + 1.89·7-s + 0.353·8-s + 0.693·9-s + 0.316·10-s + 0.767·11-s + 0.650·12-s − 0.189·13-s + 1.33·14-s + 0.581·15-s + 0.250·16-s − 0.901·17-s + 0.490·18-s + 0.316·19-s + 0.223·20-s + 2.46·21-s + 0.542·22-s − 0.284·23-s + 0.460·24-s + 0.200·25-s − 0.133·26-s − 0.399·27-s + 0.946·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.180488489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.180488489\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 - T \) |
| good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 - 5.00T + 7T^{2} \) |
| 11 | \( 1 - 2.54T + 11T^{2} \) |
| 13 | \( 1 + 0.683T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 5.59T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 8.40T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 + 9.50T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.74T + 97T^{2} \) |
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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
−8.202878577896635404849286018403, −7.34940417611477692859902115838, −6.87315358207643217932393209605, −5.69160748235867701410347958526, −5.16353155398095971808607927554, −4.24641749065735612327541124140, −3.81460017024888774764729237286, −2.66325041599321299572513498359, −2.02430692483632968580600844408, −1.41285698446133652460439211703,
1.41285698446133652460439211703, 2.02430692483632968580600844408, 2.66325041599321299572513498359, 3.81460017024888774764729237286, 4.24641749065735612327541124140, 5.16353155398095971808607927554, 5.69160748235867701410347958526, 6.87315358207643217932393209605, 7.34940417611477692859902115838, 8.202878577896635404849286018403
