| L(s) = 1 | − 3.19·5-s + 4.72·7-s − 1.62·11-s + 6.89·13-s + 7.02·17-s − 0.601·19-s + 3.69·23-s + 5.22·25-s − 7.32·29-s − 5.49·31-s − 15.1·35-s + 37-s + 0.125·41-s + 11.6·43-s + 9.32·47-s + 15.3·49-s − 6.83·53-s + 5.20·55-s + 2.92·59-s + 2·61-s − 22.0·65-s − 10.8·67-s + 9.32·71-s + 0.331·73-s − 7.69·77-s − 0.293·79-s − 9.49·83-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 1.78·7-s − 0.490·11-s + 1.91·13-s + 1.70·17-s − 0.138·19-s + 0.770·23-s + 1.04·25-s − 1.36·29-s − 0.987·31-s − 2.55·35-s + 0.164·37-s + 0.0196·41-s + 1.78·43-s + 1.36·47-s + 2.19·49-s − 0.938·53-s + 0.701·55-s + 0.381·59-s + 0.256·61-s − 2.73·65-s − 1.32·67-s + 1.10·71-s + 0.0387·73-s − 0.876·77-s − 0.0330·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.165940943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.165940943\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 + 0.601T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 41 | \( 1 - 0.125T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 9.32T + 47T^{2} \) |
| 53 | \( 1 + 6.83T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 - 0.331T + 73T^{2} \) |
| 79 | \( 1 + 0.293T + 79T^{2} \) |
| 83 | \( 1 + 9.49T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−8.035024991008844600509997774832, −7.69656863561417888274375421892, −7.09792421853667390817571699253, −5.67973470696915410447778742421, −5.43837250513351483724616601061, −4.26597366356458862133409711736, −3.90630270786154745076941752352, −3.02122996995458460991540096637, −1.63683927067672268813227848347, −0.873401799984507647109528844188,
0.873401799984507647109528844188, 1.63683927067672268813227848347, 3.02122996995458460991540096637, 3.90630270786154745076941752352, 4.26597366356458862133409711736, 5.43837250513351483724616601061, 5.67973470696915410447778742421, 7.09792421853667390817571699253, 7.69656863561417888274375421892, 8.035024991008844600509997774832
