| L(s) = 1 | − 2.15·3-s − 0.228·5-s + 2.56·7-s + 1.64·9-s − 0.376·11-s + 2.42·13-s + 0.492·15-s − 17-s − 3.73·19-s − 5.52·21-s − 1.19·23-s − 4.94·25-s + 2.92·27-s + 7.25·29-s + 8.58·31-s + 0.811·33-s − 0.585·35-s − 5.61·37-s − 5.22·39-s + 6.42·41-s + 8.49·43-s − 0.374·45-s − 0.519·47-s − 0.430·49-s + 2.15·51-s + 3.21·53-s + 0.0860·55-s + ⋯ |
| L(s) = 1 | − 1.24·3-s − 0.102·5-s + 0.968·7-s + 0.546·9-s − 0.113·11-s + 0.672·13-s + 0.127·15-s − 0.242·17-s − 0.857·19-s − 1.20·21-s − 0.249·23-s − 0.989·25-s + 0.563·27-s + 1.34·29-s + 1.54·31-s + 0.141·33-s − 0.0989·35-s − 0.923·37-s − 0.836·39-s + 1.00·41-s + 1.29·43-s − 0.0558·45-s − 0.0758·47-s − 0.0614·49-s + 0.301·51-s + 0.441·53-s + 0.0116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175398332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175398332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 0.228T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 0.376T + 11T^{2} \) |
| 13 | \( 1 - 2.42T + 13T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 1.19T + 23T^{2} \) |
| 29 | \( 1 - 7.25T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 - 6.42T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 + 0.519T + 47T^{2} \) |
| 53 | \( 1 - 3.21T + 53T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 - 4.36T + 73T^{2} \) |
| 79 | \( 1 - 3.27T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 7.74T + 89T^{2} \) |
| 97 | \( 1 + 1.60T + 97T^{2} \) |
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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
−8.345578336490716741874264938589, −7.78634130234885378475843473798, −6.77407467144151578405112050453, −6.12045692993259279970308490273, −5.60667782965421181795936872077, −4.57740370747894231659350613540, −4.31393437415153988362601463191, −2.90282231760975378710108685135, −1.74461237623775285126042267555, −0.67882651676847085878574165144,
0.67882651676847085878574165144, 1.74461237623775285126042267555, 2.90282231760975378710108685135, 4.31393437415153988362601463191, 4.57740370747894231659350613540, 5.60667782965421181795936872077, 6.12045692993259279970308490273, 6.77407467144151578405112050453, 7.78634130234885378475843473798, 8.345578336490716741874264938589
