| L(s) = 1 | − 2.32·3-s + 5-s − 4.92·7-s + 2.40·9-s + 3.75·11-s + 0.834·13-s − 2.32·15-s + 7.22·17-s − 8.52·19-s + 11.4·21-s − 0.792·23-s + 25-s + 1.37·27-s + 5.98·29-s − 10.3·31-s − 8.73·33-s − 4.92·35-s + 2.77·37-s − 1.94·39-s − 3.48·41-s − 0.259·43-s + 2.40·45-s + 3.49·47-s + 17.2·49-s − 16.7·51-s + 7.01·53-s + 3.75·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s − 1.86·7-s + 0.802·9-s + 1.13·11-s + 0.231·13-s − 0.600·15-s + 1.75·17-s − 1.95·19-s + 2.50·21-s − 0.165·23-s + 0.200·25-s + 0.264·27-s + 1.11·29-s − 1.86·31-s − 1.52·33-s − 0.832·35-s + 0.456·37-s − 0.310·39-s − 0.543·41-s − 0.0396·43-s + 0.359·45-s + 0.509·47-s + 2.46·49-s − 2.35·51-s + 0.963·53-s + 0.506·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7764377211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7764377211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.32T + 3T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 13 | \( 1 - 0.834T + 13T^{2} \) |
| 17 | \( 1 - 7.22T + 17T^{2} \) |
| 19 | \( 1 + 8.52T + 19T^{2} \) |
| 23 | \( 1 + 0.792T + 23T^{2} \) |
| 29 | \( 1 - 5.98T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 + 3.48T + 41T^{2} \) |
| 43 | \( 1 + 0.259T + 43T^{2} \) |
| 47 | \( 1 - 3.49T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 + 4.84T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 4.36T + 67T^{2} \) |
| 71 | \( 1 + 6.83T + 71T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 8.96T + 89T^{2} \) |
| 97 | \( 1 - 0.902T + 97T^{2} \) |
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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
−7.58249023622403799035630829288, −6.71719666544943981407881282477, −6.38555966466836791533340366103, −5.92949586354687931531392836070, −5.33711193805948546705047628214, −4.23910178093502889671197187517, −3.62744295322624974635647221682, −2.76842653755485339544774488185, −1.50025552854804169205295007189, −0.48457286512559605043241012521,
0.48457286512559605043241012521, 1.50025552854804169205295007189, 2.76842653755485339544774488185, 3.62744295322624974635647221682, 4.23910178093502889671197187517, 5.33711193805948546705047628214, 5.92949586354687931531392836070, 6.38555966466836791533340366103, 6.71719666544943981407881282477, 7.58249023622403799035630829288
