| L(s) = 1 | − 1.74·2-s + 1.05·4-s − 1.38·5-s − 2.70·7-s + 1.65·8-s + 2.42·10-s − 11-s − 13-s + 4.73·14-s − 4.99·16-s − 6.27·17-s + 5.86·19-s − 1.46·20-s + 1.74·22-s + 3.21·23-s − 3.07·25-s + 1.74·26-s − 2.85·28-s − 2.53·29-s − 8.95·31-s + 5.42·32-s + 10.9·34-s + 3.75·35-s + 9.76·37-s − 10.2·38-s − 2.29·40-s − 5.16·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.526·4-s − 0.620·5-s − 1.02·7-s + 0.584·8-s + 0.766·10-s − 0.301·11-s − 0.277·13-s + 1.26·14-s − 1.24·16-s − 1.52·17-s + 1.34·19-s − 0.326·20-s + 0.372·22-s + 0.669·23-s − 0.614·25-s + 0.342·26-s − 0.539·28-s − 0.470·29-s − 1.60·31-s + 0.958·32-s + 1.87·34-s + 0.635·35-s + 1.60·37-s − 1.66·38-s − 0.362·40-s − 0.806·41-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
| \(L(1)\) |
\(\approx\) |
\(0.3998509095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3998509095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.74T + 2T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 - 5.86T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 8.95T + 31T^{2} \) |
| 37 | \( 1 - 9.76T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 - 4.50T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 - 2.67T + 53T^{2} \) |
| 59 | \( 1 - 8.42T + 59T^{2} \) |
| 61 | \( 1 - 4.02T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 + 8.74T + 79T^{2} \) |
| 83 | \( 1 - 3.05T + 83T^{2} \) |
| 89 | \( 1 + 2.15T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−9.514773174875937039260286229547, −9.031744404642106067297654067471, −8.116086499055095705756955967449, −7.32581795315772374219201655253, −6.82100466225731588146912681631, −5.56764786385371872510279400251, −4.43589105510554454653117786651, −3.42483649894175697173997765034, −2.17442639291106482392965091361, −0.53848698578671575359427177065,
0.53848698578671575359427177065, 2.17442639291106482392965091361, 3.42483649894175697173997765034, 4.43589105510554454653117786651, 5.56764786385371872510279400251, 6.82100466225731588146912681631, 7.32581795315772374219201655253, 8.116086499055095705756955967449, 9.031744404642106067297654067471, 9.514773174875937039260286229547
