| L(s) = 1 | + 1.64·2-s − 3-s + 0.703·4-s − 3.28·5-s − 1.64·6-s − 2.13·8-s + 9-s − 5.39·10-s − 3.35·11-s − 0.703·12-s − 5.75·13-s + 3.28·15-s − 4.91·16-s − 6.12·17-s + 1.64·18-s − 3.14·19-s − 2.30·20-s − 5.52·22-s − 9.17·23-s + 2.13·24-s + 5.77·25-s − 9.46·26-s − 27-s + 0.986·29-s + 5.39·30-s + 4.50·31-s − 3.81·32-s + ⋯ |
| L(s) = 1 | + 1.16·2-s − 0.577·3-s + 0.351·4-s − 1.46·5-s − 0.671·6-s − 0.753·8-s + 0.333·9-s − 1.70·10-s − 1.01·11-s − 0.203·12-s − 1.59·13-s + 0.847·15-s − 1.22·16-s − 1.48·17-s + 0.387·18-s − 0.720·19-s − 0.516·20-s − 1.17·22-s − 1.91·23-s + 0.435·24-s + 1.15·25-s − 1.85·26-s − 0.192·27-s + 0.183·29-s + 0.985·30-s + 0.808·31-s − 0.673·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05807611939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05807611939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 1.64T + 2T^{2} \) |
| 5 | \( 1 + 3.28T + 5T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 + 6.12T + 17T^{2} \) |
| 19 | \( 1 + 3.14T + 19T^{2} \) |
| 23 | \( 1 + 9.17T + 23T^{2} \) |
| 29 | \( 1 - 0.986T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 2.61T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 1.80T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 + 8.44T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| 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
−7.85232506402497667667288962134, −7.37647454519760485444730257902, −6.39946272023261242992109698040, −5.92919613694997308526234615885, −4.74433623025986984064072626274, −4.58504514870922793734501438568, −4.04941382266018943498580902789, −2.91493712267258872026518317363, −2.28348303724184658616979217053, −0.099801108460767392393340632926,
0.099801108460767392393340632926, 2.28348303724184658616979217053, 2.91493712267258872026518317363, 4.04941382266018943498580902789, 4.58504514870922793734501438568, 4.74433623025986984064072626274, 5.92919613694997308526234615885, 6.39946272023261242992109698040, 7.37647454519760485444730257902, 7.85232506402497667667288962134
