| L(s) = 1 | + 2.14·2-s + 2.61·4-s − 2.26·5-s − 3.68·7-s + 1.32·8-s − 4.85·10-s + 11-s + 13-s − 7.91·14-s − 2.38·16-s − 0.851·17-s − 5.11·19-s − 5.91·20-s + 2.14·22-s − 5.05·23-s + 0.112·25-s + 2.14·26-s − 9.64·28-s − 5.42·29-s + 5.28·31-s − 7.77·32-s − 1.82·34-s + 8.32·35-s + 5.65·37-s − 11.0·38-s − 3.00·40-s − 1.36·41-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 1.30·4-s − 1.01·5-s − 1.39·7-s + 0.469·8-s − 1.53·10-s + 0.301·11-s + 0.277·13-s − 2.11·14-s − 0.595·16-s − 0.206·17-s − 1.17·19-s − 1.32·20-s + 0.458·22-s − 1.05·23-s + 0.0224·25-s + 0.421·26-s − 1.82·28-s − 1.00·29-s + 0.949·31-s − 1.37·32-s − 0.313·34-s + 1.40·35-s + 0.930·37-s − 1.78·38-s − 0.474·40-s − 0.212·41-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 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 17 | \( 1 + 0.851T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + 5.42T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 + 0.290T + 47T^{2} \) |
| 53 | \( 1 + 9.56T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 - 0.795T + 61T^{2} \) |
| 67 | \( 1 - 2.54T + 67T^{2} \) |
| 71 | \( 1 - 0.594T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 7.33T + 83T^{2} \) |
| 89 | \( 1 + 4.71T + 89T^{2} \) |
| 97 | \( 1 - 9.98T + 97T^{2} \) |
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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
−9.295103080628277737416734942131, −8.331419695108709090063198039851, −7.33693556313555836208885901882, −6.34291821252941680921869986515, −6.09588602069507624669087513532, −4.74484880494961749565612121021, −3.91357912141063676385445154013, −3.48398407706382139845738275466, −2.35427042822991324631957814515, 0,
2.35427042822991324631957814515, 3.48398407706382139845738275466, 3.91357912141063676385445154013, 4.74484880494961749565612121021, 6.09588602069507624669087513532, 6.34291821252941680921869986515, 7.33693556313555836208885901882, 8.331419695108709090063198039851, 9.295103080628277737416734942131
