| L(s) = 1 | + 2.73·2-s + 0.521·3-s + 5.48·4-s + 2.58·5-s + 1.42·6-s − 4.63·7-s + 9.54·8-s − 2.72·9-s + 7.08·10-s + 0.185·11-s + 2.86·12-s − 0.541·13-s − 12.6·14-s + 1.34·15-s + 15.1·16-s − 17-s − 7.46·18-s + 2.05·19-s + 14.2·20-s − 2.41·21-s + 0.507·22-s − 5.10·23-s + 4.97·24-s + 1.69·25-s − 1.48·26-s − 2.98·27-s − 25.4·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.300·3-s + 2.74·4-s + 1.15·5-s + 0.582·6-s − 1.75·7-s + 3.37·8-s − 0.909·9-s + 2.23·10-s + 0.0559·11-s + 0.826·12-s − 0.150·13-s − 3.39·14-s + 0.348·15-s + 3.78·16-s − 0.242·17-s − 1.75·18-s + 0.471·19-s + 3.17·20-s − 0.527·21-s + 0.108·22-s − 1.06·23-s + 1.01·24-s + 0.339·25-s − 0.290·26-s − 0.574·27-s − 4.80·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.185088895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.185088895\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 3 | \( 1 - 0.521T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 4.63T + 7T^{2} \) |
| 11 | \( 1 - 0.185T + 11T^{2} \) |
| 13 | \( 1 + 0.541T + 13T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 5.06T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 + 7.13T + 37T^{2} \) |
| 41 | \( 1 + 5.24T + 41T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.13T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 2.16T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 3.69T + 79T^{2} \) |
| 83 | \( 1 - 0.638T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 0.184T + 97T^{2} \) |
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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
−10.31765687089569110974783820959, −9.946585857784160530039428635529, −8.702619547321937031005736971773, −7.23413341988765190442944887076, −6.33148521942596784475817875171, −5.97628678345225876747934506833, −5.08924240694536312180055521061, −3.71497106260160450938434602622, −2.97515138498380124130015116612, −2.15500107042794229661062374471,
2.15500107042794229661062374471, 2.97515138498380124130015116612, 3.71497106260160450938434602622, 5.08924240694536312180055521061, 5.97628678345225876747934506833, 6.33148521942596784475817875171, 7.23413341988765190442944887076, 8.702619547321937031005736971773, 9.946585857784160530039428635529, 10.31765687089569110974783820959
