| L(s) = 1 | + 2.44·3-s + 4.02·7-s + 2.98·9-s + 1.17·11-s − 5.60·13-s + 3.00·17-s + 1.62·19-s + 9.85·21-s + 0.296·23-s − 0.0323·27-s + 0.296·29-s − 1.77·31-s + 2.88·33-s + 37-s − 13.7·39-s − 1.17·41-s + 5.07·43-s + 4.94·47-s + 9.23·49-s + 7.35·51-s + 8.75·53-s + 3.97·57-s + 6.72·59-s − 8.22·61-s + 12.0·63-s + 13.5·67-s + 0.724·69-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 1.52·7-s + 0.995·9-s + 0.355·11-s − 1.55·13-s + 0.728·17-s + 0.372·19-s + 2.15·21-s + 0.0617·23-s − 0.00623·27-s + 0.0549·29-s − 0.318·31-s + 0.501·33-s + 0.164·37-s − 2.19·39-s − 0.183·41-s + 0.774·43-s + 0.721·47-s + 1.31·49-s + 1.02·51-s + 1.20·53-s + 0.526·57-s + 0.875·59-s − 1.05·61-s + 1.51·63-s + 1.65·67-s + 0.0872·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.337254573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.337254573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.296T + 23T^{2} \) |
| 29 | \( 1 - 0.296T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 41 | \( 1 + 1.17T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 2.74T + 73T^{2} \) |
| 79 | \( 1 - 6.18T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 5.13T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
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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
−7.932151728890781640466420780481, −7.49406334137132507143948353913, −6.85392526588682388748998258876, −5.54933261581981607513679407187, −5.03375787463070637348106844674, −4.22372894999676965463650247724, −3.52191059093501372606285901241, −2.50250351296475402773093059131, −2.08096665147691937063428265351, −1.02586903372623319318947606146,
1.02586903372623319318947606146, 2.08096665147691937063428265351, 2.50250351296475402773093059131, 3.52191059093501372606285901241, 4.22372894999676965463650247724, 5.03375787463070637348106844674, 5.54933261581981607513679407187, 6.85392526588682388748998258876, 7.49406334137132507143948353913, 7.932151728890781640466420780481
