| L(s) = 1 | − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s + 1.53·11-s − 1.87·13-s − 16-s + 1.53·17-s − 18-s − 1.53·22-s + 0.347·23-s + 1.34·25-s + 1.87·26-s − 31-s − 1.53·34-s + 1.53·40-s − 1.87·41-s − 1.87·43-s + 1.53·45-s − 0.347·46-s − 1.87·47-s + 49-s − 1.34·50-s + 0.347·53-s + 2.34·55-s + 62-s + ⋯ |
| L(s) = 1 | − 2-s + 1.53·5-s + 8-s + 9-s − 1.53·10-s + 1.53·11-s − 1.87·13-s − 16-s + 1.53·17-s − 18-s − 1.53·22-s + 0.347·23-s + 1.34·25-s + 1.87·26-s − 31-s − 1.53·34-s + 1.53·40-s − 1.87·41-s − 1.87·43-s + 1.53·45-s − 0.347·46-s − 1.87·47-s + 49-s − 1.34·50-s + 0.347·53-s + 2.34·55-s + 62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9302675303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9302675303\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1759 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.53T + T^{2} \) |
| 13 | \( 1 + 1.87T + T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 0.347T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 + 1.87T + T^{2} \) |
| 47 | \( 1 + 1.87T + T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.347T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.594561576886732151215276120247, −9.061196552208389652710906885188, −8.007014336265295763649067814375, −7.08214067452355679434743392122, −6.64653186810739069561181595495, −5.33970252119321555311650545523, −4.78912658868857599530870358859, −3.51122826036390459186817033217, −1.95418773147952980826953839856, −1.36010945604830553381911373416,
1.36010945604830553381911373416, 1.95418773147952980826953839856, 3.51122826036390459186817033217, 4.78912658868857599530870358859, 5.33970252119321555311650545523, 6.64653186810739069561181595495, 7.08214067452355679434743392122, 8.007014336265295763649067814375, 9.061196552208389652710906885188, 9.594561576886732151215276120247
