| L(s) = 1 | − 0.0772·2-s − 3-s − 1.99·4-s + 3.97·5-s + 0.0772·6-s + 0.969·7-s + 0.308·8-s + 9-s − 0.306·10-s + 2.24·11-s + 1.99·12-s + 4.20·13-s − 0.0748·14-s − 3.97·15-s + 3.96·16-s − 17-s − 0.0772·18-s + 5.15·19-s − 7.92·20-s − 0.969·21-s − 0.173·22-s + 0.911·23-s − 0.308·24-s + 10.7·25-s − 0.324·26-s − 27-s − 1.93·28-s + ⋯ |
| L(s) = 1 | − 0.0546·2-s − 0.577·3-s − 0.997·4-s + 1.77·5-s + 0.0315·6-s + 0.366·7-s + 0.109·8-s + 0.333·9-s − 0.0970·10-s + 0.676·11-s + 0.575·12-s + 1.16·13-s − 0.0200·14-s − 1.02·15-s + 0.991·16-s − 0.242·17-s − 0.0182·18-s + 1.18·19-s − 1.77·20-s − 0.211·21-s − 0.0369·22-s + 0.189·23-s − 0.0629·24-s + 2.15·25-s − 0.0636·26-s − 0.192·27-s − 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213154353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213154353\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0772T + 2T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 - 0.969T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 4.20T + 13T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 0.911T + 23T^{2} \) |
| 29 | \( 1 - 8.81T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 - 6.28T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 7.12T + 59T^{2} \) |
| 61 | \( 1 + 7.64T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 0.523T + 71T^{2} \) |
| 73 | \( 1 + 9.82T + 73T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.691515450714350350629889193752, −7.80445035804982611238494032939, −6.66512571819539547699275226923, −6.14719516795873069253144472674, −5.43681448137698381946919500933, −4.92609121977346670628055455354, −4.00568193399757407858600371725, −2.95126831512056726536120696296, −1.57729895695523216901888319583, −1.03336612488968903161529614059,
1.03336612488968903161529614059, 1.57729895695523216901888319583, 2.95126831512056726536120696296, 4.00568193399757407858600371725, 4.92609121977346670628055455354, 5.43681448137698381946919500933, 6.14719516795873069253144472674, 6.66512571819539547699275226923, 7.80445035804982611238494032939, 8.691515450714350350629889193752
