| L(s) = 1 | + 2.17·3-s − 1.55·5-s + 3.67·7-s + 1.71·9-s − 6.34·11-s − 4.15·13-s − 3.37·15-s + 17-s + 4.98·19-s + 7.98·21-s − 3.19·23-s − 2.57·25-s − 2.78·27-s − 5.70·29-s − 0.739·31-s − 13.7·33-s − 5.71·35-s + 3.82·37-s − 9.01·39-s − 10.3·41-s + 2.00·43-s − 2.67·45-s − 12.9·47-s + 6.50·49-s + 2.17·51-s + 1.42·53-s + 9.87·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 0.695·5-s + 1.38·7-s + 0.572·9-s − 1.91·11-s − 1.15·13-s − 0.872·15-s + 0.242·17-s + 1.14·19-s + 1.74·21-s − 0.665·23-s − 0.515·25-s − 0.535·27-s − 1.05·29-s − 0.132·31-s − 2.40·33-s − 0.966·35-s + 0.629·37-s − 1.44·39-s − 1.61·41-s + 0.305·43-s − 0.398·45-s − 1.88·47-s + 0.928·49-s + 0.304·51-s + 0.195·53-s + 1.33·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 + 1.55T + 5T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 0.739T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 61 | \( 1 - 7.05T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 9.68T + 73T^{2} \) |
| 79 | \( 1 + 3.90T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.071605514659665108783387415602, −7.70805050391611537470476017047, −7.10724692468384184247237774158, −5.41319240830051373149195844984, −5.17068419771348568612386126695, −4.17568858118590229286203369702, −3.28177302136189864483395946810, −2.49842014425133489811508590603, −1.78001730447424261741527438755, 0,
1.78001730447424261741527438755, 2.49842014425133489811508590603, 3.28177302136189864483395946810, 4.17568858118590229286203369702, 5.17068419771348568612386126695, 5.41319240830051373149195844984, 7.10724692468384184247237774158, 7.70805050391611537470476017047, 8.071605514659665108783387415602
