| L(s) = 1 | + 2-s − 3-s + 4-s + 3.66·5-s − 6-s − 2.09·7-s + 8-s + 9-s + 3.66·10-s − 4.52·11-s − 12-s + 3.23·13-s − 2.09·14-s − 3.66·15-s + 16-s + 4.80·17-s + 18-s + 19-s + 3.66·20-s + 2.09·21-s − 4.52·22-s + 1.12·23-s − 24-s + 8.42·25-s + 3.23·26-s − 27-s − 2.09·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.63·5-s − 0.408·6-s − 0.793·7-s + 0.353·8-s + 0.333·9-s + 1.15·10-s − 1.36·11-s − 0.288·12-s + 0.897·13-s − 0.560·14-s − 0.946·15-s + 0.250·16-s + 1.16·17-s + 0.235·18-s + 0.229·19-s + 0.819·20-s + 0.457·21-s − 0.965·22-s + 0.234·23-s − 0.204·24-s + 1.68·25-s + 0.634·26-s − 0.192·27-s − 0.396·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.433599432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.433599432\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 + 2.09T + 7T^{2} \) |
| 11 | \( 1 + 4.52T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 23 | \( 1 - 1.12T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 - 5.56T + 41T^{2} \) |
| 43 | \( 1 - 6.72T + 43T^{2} \) |
| 47 | \( 1 - 1.59T + 47T^{2} \) |
| 59 | \( 1 + 1.40T + 59T^{2} \) |
| 61 | \( 1 + 1.83T + 61T^{2} \) |
| 67 | \( 1 + 8.99T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.50T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 7.68T + 89T^{2} \) |
| 97 | \( 1 - 5.02T + 97T^{2} \) |
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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
−7.87964833764979176126418751057, −7.13792873442021189117796371970, −6.29314157306319592604530057921, −5.84747028452978098465083004179, −5.43923125475037039131974311808, −4.72026607816879875567095409968, −3.52120505463010978622734404378, −2.83784381233275323002547280165, −1.98185261877936894811218700321, −0.919912457353224874870658628106,
0.919912457353224874870658628106, 1.98185261877936894811218700321, 2.83784381233275323002547280165, 3.52120505463010978622734404378, 4.72026607816879875567095409968, 5.43923125475037039131974311808, 5.84747028452978098465083004179, 6.29314157306319592604530057921, 7.13792873442021189117796371970, 7.87964833764979176126418751057
