| L(s) = 1 | − 3-s − 2·4-s − 5-s + 3·7-s + 9-s − 3·11-s + 2·12-s + 15-s + 4·16-s − 3·17-s + 2·20-s − 3·21-s + 3·23-s + 25-s − 27-s − 6·28-s − 6·29-s + 6·31-s + 3·33-s − 3·35-s − 2·36-s − 9·37-s − 3·41-s + 10·43-s + 6·44-s − 45-s − 12·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 0.447·20-s − 0.654·21-s + 0.625·23-s + 1/5·25-s − 0.192·27-s − 1.13·28-s − 1.11·29-s + 1.07·31-s + 0.522·33-s − 0.507·35-s − 1/3·36-s − 1.47·37-s − 0.468·41-s + 1.52·43-s + 0.904·44-s − 0.149·45-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8892445123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8892445123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−8.765154230348609914473829791484, −8.162800629217765674167583246921, −7.56669985510585965966341183018, −6.61462902511927818987493434872, −5.44114193130814224462783205875, −4.98904539167326410769878999374, −4.36403452129879769416159437543, −3.38037129798030626259220459654, −1.95733443232044255651478454976, −0.61990195475778641136919885683,
0.61990195475778641136919885683, 1.95733443232044255651478454976, 3.38037129798030626259220459654, 4.36403452129879769416159437543, 4.98904539167326410769878999374, 5.44114193130814224462783205875, 6.61462902511927818987493434872, 7.56669985510585965966341183018, 8.162800629217765674167583246921, 8.765154230348609914473829791484
