| L(s) = 1 | − 2-s − 1.15·3-s + 4-s − 2.65·5-s + 1.15·6-s + 4.05·7-s − 8-s − 1.65·9-s + 2.65·10-s − 11-s − 1.15·12-s + 3.70·13-s − 4.05·14-s + 3.07·15-s + 16-s + 5.32·17-s + 1.65·18-s + 8.24·19-s − 2.65·20-s − 4.70·21-s + 22-s + 2.73·23-s + 1.15·24-s + 2.04·25-s − 3.70·26-s + 5.39·27-s + 4.05·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.669·3-s + 0.5·4-s − 1.18·5-s + 0.473·6-s + 1.53·7-s − 0.353·8-s − 0.552·9-s + 0.839·10-s − 0.301·11-s − 0.334·12-s + 1.02·13-s − 1.08·14-s + 0.794·15-s + 0.250·16-s + 1.29·17-s + 0.390·18-s + 1.89·19-s − 0.593·20-s − 1.02·21-s + 0.213·22-s + 0.570·23-s + 0.236·24-s + 0.409·25-s − 0.726·26-s + 1.03·27-s + 0.766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.057601381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057601381\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 - 5.32T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 23 | \( 1 - 2.73T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 5.17T + 31T^{2} \) |
| 37 | \( 1 - 5.48T + 37T^{2} \) |
| 41 | \( 1 + 0.230T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 9.71T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 0.246T + 83T^{2} \) |
| 89 | \( 1 - 0.350T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−8.152923665519346363418242500514, −7.70214414255343082621490256472, −7.38565763765760984914000227756, −6.07741919118436293815336747078, −5.44874566789856785258648685333, −4.82372876573724063163107587126, −3.71377395170733388970256151773, −2.98531856411388498293889449251, −1.48558771070187194359584178029, −0.73709708403476452463107083064,
0.73709708403476452463107083064, 1.48558771070187194359584178029, 2.98531856411388498293889449251, 3.71377395170733388970256151773, 4.82372876573724063163107587126, 5.44874566789856785258648685333, 6.07741919118436293815336747078, 7.38565763765760984914000227756, 7.70214414255343082621490256472, 8.152923665519346363418242500514
