| L(s) = 1 | − 2-s + 4-s + 0.454·5-s − 8-s − 0.454·10-s − 11-s − 0.909·13-s + 16-s + 17-s − 3.88·19-s + 0.454·20-s + 22-s − 8.33·23-s − 4.79·25-s + 0.909·26-s + 2.79·29-s + 9.58·31-s − 32-s − 34-s + 7.88·37-s + 3.88·38-s − 0.454·40-s − 1.79·41-s + 7.88·43-s − 44-s + 8.33·46-s + 0.338·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.203·5-s − 0.353·8-s − 0.143·10-s − 0.301·11-s − 0.252·13-s + 0.250·16-s + 0.242·17-s − 0.890·19-s + 0.101·20-s + 0.213·22-s − 1.73·23-s − 0.958·25-s + 0.178·26-s + 0.518·29-s + 1.72·31-s − 0.176·32-s − 0.171·34-s + 1.29·37-s + 0.629·38-s − 0.0719·40-s − 0.280·41-s + 1.20·43-s − 0.150·44-s + 1.22·46-s + 0.0493·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 0.454T + 5T^{2} \) |
| 13 | \( 1 + 0.909T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + 3.88T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 9.58T + 31T^{2} \) |
| 37 | \( 1 - 7.88T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 7.88T + 43T^{2} \) |
| 47 | \( 1 - 0.338T + 47T^{2} \) |
| 53 | \( 1 + 0.909T + 53T^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 3.79T + 67T^{2} \) |
| 71 | \( 1 - 4.79T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.36T + 79T^{2} \) |
| 83 | \( 1 - 1.97T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−7.49713627177369537235960462240, −6.65249748573987834860987861573, −6.09094708357095890521726670601, −5.50455656188135923755759340654, −4.43627719664045514170067265780, −3.88914147764201965129019625467, −2.65331529557622041594585174775, −2.23684082460638736117312732227, −1.12335487363504112757685226243, 0,
1.12335487363504112757685226243, 2.23684082460638736117312732227, 2.65331529557622041594585174775, 3.88914147764201965129019625467, 4.43627719664045514170067265780, 5.50455656188135923755759340654, 6.09094708357095890521726670601, 6.65249748573987834860987861573, 7.49713627177369537235960462240
