| L(s) = 1 | + 1.23·5-s − 4.82·7-s + 5.94·11-s + 2·13-s + 7.17·17-s − 2.82·19-s + 1.23·23-s − 3.48·25-s + 7.17·29-s − 1.17·31-s − 5.94·35-s − 37-s − 8.40·41-s − 2.82·43-s + 8.40·47-s + 16.3·49-s − 5.94·53-s + 7.31·55-s − 4.71·59-s − 3.65·61-s + 2.46·65-s − 3.17·67-s − 3.48·71-s + 9.17·73-s − 28.6·77-s − 8.48·79-s + 9.42·83-s + ⋯ |
| L(s) = 1 | + 0.550·5-s − 1.82·7-s + 1.79·11-s + 0.554·13-s + 1.73·17-s − 0.648·19-s + 0.256·23-s − 0.697·25-s + 1.33·29-s − 0.210·31-s − 1.00·35-s − 0.164·37-s − 1.31·41-s − 0.431·43-s + 1.22·47-s + 2.33·49-s − 0.816·53-s + 0.986·55-s − 0.613·59-s − 0.468·61-s + 0.305·65-s − 0.387·67-s − 0.413·71-s + 1.07·73-s − 3.26·77-s − 0.954·79-s + 1.03·83-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
| \(L(1)\) |
\(\approx\) |
\(2.054440274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.054440274\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 5.94T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.17T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 29 | \( 1 - 7.17T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 3.17T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 - 9.17T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−8.321402106785139870473353149731, −7.26929997218385697023580350268, −6.51295198582333177580394915435, −6.23188400781860728952860685549, −5.56037265450971955436547859215, −4.33784772265256096138434633028, −3.49656547979114258580612910636, −3.12937470512590588794776083798, −1.78800299782540312422183563415, −0.798817168927063699736813848234,
0.798817168927063699736813848234, 1.78800299782540312422183563415, 3.12937470512590588794776083798, 3.49656547979114258580612910636, 4.33784772265256096138434633028, 5.56037265450971955436547859215, 6.23188400781860728952860685549, 6.51295198582333177580394915435, 7.26929997218385697023580350268, 8.321402106785139870473353149731
