| L(s) = 1 | − 0.311·2-s − 1.90·4-s − 2.21·5-s + 1.52·7-s + 1.21·8-s + 0.688·10-s − 11-s + 13-s − 0.474·14-s + 3.42·16-s + 1.73·17-s + 4.28·19-s + 4.21·20-s + 0.311·22-s − 0.474·23-s − 0.0967·25-s − 0.311·26-s − 2.90·28-s − 5.11·29-s − 4.21·31-s − 3.49·32-s − 0.541·34-s − 3.37·35-s + 0.428·37-s − 1.33·38-s − 2.68·40-s − 6.28·41-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.951·4-s − 0.990·5-s + 0.576·7-s + 0.429·8-s + 0.217·10-s − 0.301·11-s + 0.277·13-s − 0.126·14-s + 0.857·16-s + 0.421·17-s + 0.982·19-s + 0.942·20-s + 0.0663·22-s − 0.0989·23-s − 0.0193·25-s − 0.0610·26-s − 0.548·28-s − 0.950·29-s − 0.756·31-s − 0.617·32-s − 0.0928·34-s − 0.570·35-s + 0.0704·37-s − 0.216·38-s − 0.425·40-s − 0.980·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 - 4.28T + 19T^{2} \) |
| 23 | \( 1 + 0.474T + 23T^{2} \) |
| 29 | \( 1 + 5.11T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 0.428T + 37T^{2} \) |
| 41 | \( 1 + 6.28T + 41T^{2} \) |
| 43 | \( 1 + 8.16T + 43T^{2} \) |
| 47 | \( 1 + 5.05T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 + 5.80T + 61T^{2} \) |
| 67 | \( 1 + 6.02T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 + 8.08T + 73T^{2} \) |
| 79 | \( 1 - 0.260T + 79T^{2} \) |
| 83 | \( 1 - 1.37T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 - 0.428T + 97T^{2} \) |
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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
−9.187347422496321901645503105190, −8.327996704273390548937953473693, −7.83978373234788597112221702987, −7.11540925995756963998457129605, −5.64410797360302589870720682825, −4.97682812301428659657154848241, −4.01277793739973388350372257730, −3.29499333180796031086062307863, −1.48193585247206541436791240905, 0,
1.48193585247206541436791240905, 3.29499333180796031086062307863, 4.01277793739973388350372257730, 4.97682812301428659657154848241, 5.64410797360302589870720682825, 7.11540925995756963998457129605, 7.83978373234788597112221702987, 8.327996704273390548937953473693, 9.187347422496321901645503105190
