| L(s) = 1 | − 2-s − 3.14·3-s + 4-s + 4.25·5-s + 3.14·6-s − 0.789·7-s − 8-s + 6.86·9-s − 4.25·10-s − 11-s − 3.14·12-s + 2.86·13-s + 0.789·14-s − 13.3·15-s + 16-s − 4.87·17-s − 6.86·18-s + 4.25·20-s + 2.47·21-s + 22-s − 3.00·23-s + 3.14·24-s + 13.0·25-s − 2.86·26-s − 12.1·27-s − 0.789·28-s − 6.79·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.81·3-s + 0.5·4-s + 1.90·5-s + 1.28·6-s − 0.298·7-s − 0.353·8-s + 2.28·9-s − 1.34·10-s − 0.301·11-s − 0.906·12-s + 0.794·13-s + 0.210·14-s − 3.44·15-s + 0.250·16-s − 1.18·17-s − 1.61·18-s + 0.951·20-s + 0.540·21-s + 0.213·22-s − 0.626·23-s + 0.641·24-s + 2.61·25-s − 0.561·26-s − 2.33·27-s − 0.149·28-s − 1.26·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8188819397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8188819397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 5 | \( 1 - 4.25T + 5T^{2} \) |
| 7 | \( 1 + 0.789T + 7T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 + 4.87T + 17T^{2} \) |
| 23 | \( 1 + 3.00T + 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 0.927T + 37T^{2} \) |
| 41 | \( 1 - 0.0787T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 1.10T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 - 0.0387T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 9.08T + 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 0.532T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.65213582155351393843845451286, −6.74450357131049643565887229828, −6.47087158225982660134182386435, −5.79505650761985050421250646822, −5.44887056516418223090616499977, −4.65849620764950555311666894725, −3.50567311785241723492275879703, −2.05057034475861913444682269827, −1.72306360333362156119760986899, −0.54889711795508969799713204302,
0.54889711795508969799713204302, 1.72306360333362156119760986899, 2.05057034475861913444682269827, 3.50567311785241723492275879703, 4.65849620764950555311666894725, 5.44887056516418223090616499977, 5.79505650761985050421250646822, 6.47087158225982660134182386435, 6.74450357131049643565887229828, 7.65213582155351393843845451286
