| L(s) = 1 | + 3-s + 4.08·5-s − 7-s + 9-s + 2.08·11-s − 2.51·13-s + 4.08·15-s − 17-s + 0.786·19-s − 21-s − 0.786·23-s + 11.6·25-s + 27-s + 6·29-s − 1.57·31-s + 2.08·33-s − 4.08·35-s + 3.29·37-s − 2.51·39-s + 2.78·41-s − 4.51·43-s + 4.08·45-s + 1.57·47-s + 49-s − 51-s + 2.27·53-s + 8.51·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.82·5-s − 0.377·7-s + 0.333·9-s + 0.628·11-s − 0.696·13-s + 1.05·15-s − 0.242·17-s + 0.180·19-s − 0.218·21-s − 0.164·23-s + 2.33·25-s + 0.192·27-s + 1.11·29-s − 0.282·31-s + 0.362·33-s − 0.690·35-s + 0.542·37-s − 0.401·39-s + 0.435·41-s − 0.687·43-s + 0.608·45-s + 0.229·47-s + 0.142·49-s − 0.140·51-s + 0.312·53-s + 1.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.683757467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.683757467\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 4.08T + 5T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 + 2.51T + 13T^{2} \) |
| 19 | \( 1 - 0.786T + 19T^{2} \) |
| 23 | \( 1 + 0.786T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 - 3.29T + 37T^{2} \) |
| 41 | \( 1 - 2.78T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 - 2.27T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 0.276T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 8.87T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 4.14T + 97T^{2} \) |
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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
−8.254633778067754851298749343696, −7.30476137278053998392216777464, −6.56571237687986116183481365468, −6.14219857620925960109007287392, −5.25893586186666592958958697068, −4.58563297673387942283511945309, −3.49403360533779075565944040734, −2.59589063651521402290307343581, −2.04729609703318455939331462559, −1.04084558349495300501524480964,
1.04084558349495300501524480964, 2.04729609703318455939331462559, 2.59589063651521402290307343581, 3.49403360533779075565944040734, 4.58563297673387942283511945309, 5.25893586186666592958958697068, 6.14219857620925960109007287392, 6.56571237687986116183481365468, 7.30476137278053998392216777464, 8.254633778067754851298749343696
