| L(s) = 1 | + 23.6·3-s − 20.0·5-s + 315.·9-s + 58.7·11-s + 691.·13-s − 473.·15-s + 526.·17-s − 127.·19-s + 2.26e3·23-s − 2.72e3·25-s + 1.72e3·27-s + 8.63e3·29-s − 1.02e4·31-s + 1.38e3·33-s − 4.72e3·37-s + 1.63e4·39-s + 6.67e3·41-s + 2.29e4·43-s − 6.33e3·45-s + 2.41e4·47-s + 1.24e4·51-s − 8.24e3·53-s − 1.17e3·55-s − 3.01e3·57-s + 3.61e4·59-s + 1.56e4·61-s − 1.38e4·65-s + ⋯ |
| L(s) = 1 | + 1.51·3-s − 0.358·5-s + 1.30·9-s + 0.146·11-s + 1.13·13-s − 0.543·15-s + 0.442·17-s − 0.0810·19-s + 0.894·23-s − 0.871·25-s + 0.455·27-s + 1.90·29-s − 1.92·31-s + 0.222·33-s − 0.567·37-s + 1.72·39-s + 0.620·41-s + 1.89·43-s − 0.466·45-s + 1.59·47-s + 0.670·51-s − 0.403·53-s − 0.0524·55-s − 0.122·57-s + 1.35·59-s + 0.538·61-s − 0.406·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.911680366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.911680366\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 23.6T + 243T^{2} \) |
| 5 | \( 1 + 20.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 58.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 691.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 526.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 127.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.26e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.67e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.41e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.24e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.02e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.20e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.57e5T + 8.58e9T^{2} \) |
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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
−10.38249417878532952391208325793, −9.249597936802522547211262931427, −8.699907419882534664655363107081, −7.86979124021983328743370528867, −7.03528242331202028221711129175, −5.69889160961546764325889164486, −4.14856140861209892071134915602, −3.43635524669646216358541566465, −2.35305302773403229718791577789, −1.03804926150087619778510313355,
1.03804926150087619778510313355, 2.35305302773403229718791577789, 3.43635524669646216358541566465, 4.14856140861209892071134915602, 5.69889160961546764325889164486, 7.03528242331202028221711129175, 7.86979124021983328743370528867, 8.699907419882534664655363107081, 9.249597936802522547211262931427, 10.38249417878532952391208325793
