| L(s) = 1 | − 2.09·2-s − 3.61·4-s − 6.99·5-s − 7·7-s + 24.3·8-s + 14.6·10-s − 11·11-s + 13.5·13-s + 14.6·14-s − 21.9·16-s − 93.3·17-s − 54.1·19-s + 25.2·20-s + 23.0·22-s + 108.·23-s − 76.0·25-s − 28.4·26-s + 25.3·28-s − 68.0·29-s − 203.·31-s − 148.·32-s + 195.·34-s + 48.9·35-s − 288.·37-s + 113.·38-s − 170.·40-s + 22.3·41-s + ⋯ |
| L(s) = 1 | − 0.740·2-s − 0.452·4-s − 0.625·5-s − 0.377·7-s + 1.07·8-s + 0.462·10-s − 0.301·11-s + 0.289·13-s + 0.279·14-s − 0.343·16-s − 1.33·17-s − 0.654·19-s + 0.282·20-s + 0.223·22-s + 0.986·23-s − 0.608·25-s − 0.214·26-s + 0.170·28-s − 0.435·29-s − 1.18·31-s − 0.820·32-s + 0.985·34-s + 0.236·35-s − 1.28·37-s + 0.484·38-s − 0.672·40-s + 0.0850·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4785051551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4785051551\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 2.09T + 8T^{2} \) |
| 5 | \( 1 + 6.99T + 125T^{2} \) |
| 13 | \( 1 - 13.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 68.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 433.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 592.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 819.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 961.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 599.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 239.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 185.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 466.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 184.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−10.03428336117601136021234954210, −8.903437289975151548488278369923, −8.677930439819020337606741027852, −7.52742574975794614219091673349, −6.85943820949264787986348420367, −5.51468090786090778624528452353, −4.43093904311205281201945385073, −3.58990683585701224574620722288, −2.00262649901860926706996519080, −0.43722935134583411379427321766,
0.43722935134583411379427321766, 2.00262649901860926706996519080, 3.58990683585701224574620722288, 4.43093904311205281201945385073, 5.51468090786090778624528452353, 6.85943820949264787986348420367, 7.52742574975794614219091673349, 8.677930439819020337606741027852, 8.903437289975151548488278369923, 10.03428336117601136021234954210
