L(s) = 1 | + 2.19·2-s − 0.364·3-s + 2.83·4-s + 2·5-s − 0.801·6-s + 7-s + 1.83·8-s − 2.86·9-s + 4.39·10-s − 2·11-s − 1.03·12-s + 1.16·13-s + 2.19·14-s − 0.728·15-s − 1.63·16-s − 1.46·17-s − 6.30·18-s + 6.19·19-s + 5.66·20-s − 0.364·21-s − 4.39·22-s − 3.63·23-s − 0.668·24-s − 25-s + 2.56·26-s + 2.13·27-s + 2.83·28-s + ⋯ |
L(s) = 1 | + 1.55·2-s − 0.210·3-s + 1.41·4-s + 0.894·5-s − 0.327·6-s + 0.377·7-s + 0.648·8-s − 0.955·9-s + 1.39·10-s − 0.603·11-s − 0.298·12-s + 0.323·13-s + 0.587·14-s − 0.188·15-s − 0.408·16-s − 0.356·17-s − 1.48·18-s + 1.42·19-s + 1.26·20-s − 0.0795·21-s − 0.937·22-s − 0.758·23-s − 0.136·24-s − 0.200·25-s + 0.502·26-s + 0.411·27-s + 0.535·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.802850876\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.802850876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 + 0.364T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 + 3.63T + 23T^{2} \) |
| 29 | \( 1 + 0.397T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 - 8.06T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6.06T + 61T^{2} \) |
| 67 | \( 1 + 4.06T + 67T^{2} \) |
| 71 | \( 1 + 0.873T + 71T^{2} \) |
| 73 | \( 1 + 4.93T + 73T^{2} \) |
| 79 | \( 1 - 4.72T + 79T^{2} \) |
| 83 | \( 1 - 6.79T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 5.23T + 97T^{2} \) |
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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
−11.80393746315499152640211260810, −11.36627952828158788410824998678, −10.17660485402033680077943599076, −9.018253079776313969254792251795, −7.70916469801081421821361618552, −6.30380849561167284619038461067, −5.63008417875851794352580007168, −4.89830623028647561220944934343, −3.42657962335328568272135699477, −2.23430470940550297170652716616,
2.23430470940550297170652716616, 3.42657962335328568272135699477, 4.89830623028647561220944934343, 5.63008417875851794352580007168, 6.30380849561167284619038461067, 7.70916469801081421821361618552, 9.018253079776313969254792251795, 10.17660485402033680077943599076, 11.36627952828158788410824998678, 11.80393746315499152640211260810