| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 3.23·5-s − 1.23·7-s − 2.23·8-s + 5.23·10-s + 0.763·11-s + 3·13-s − 2.00·14-s − 4.85·16-s − 5.23·17-s − 2·19-s + 2.00·20-s + 1.23·22-s − 23-s + 5.47·25-s + 4.85·26-s − 0.763·28-s + 3·29-s − 6.70·31-s − 3.38·32-s − 8.47·34-s − 4.00·35-s + 3.23·37-s − 3.23·38-s − 7.23·40-s − 5.47·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 1.44·5-s − 0.467·7-s − 0.790·8-s + 1.65·10-s + 0.230·11-s + 0.832·13-s − 0.534·14-s − 1.21·16-s − 1.26·17-s − 0.458·19-s + 0.447·20-s + 0.263·22-s − 0.208·23-s + 1.09·25-s + 0.951·26-s − 0.144·28-s + 0.557·29-s − 1.20·31-s − 0.597·32-s − 1.45·34-s − 0.676·35-s + 0.532·37-s − 0.524·38-s − 1.14·40-s − 0.854·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148394057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148394057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 6.94T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 4.29T + 97T^{2} \) |
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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
−12.94959674760005301372362375138, −11.64390399044447017800422711859, −10.51701510715389623659238517197, −9.418352849880820091555957559356, −8.694912320601476106960796099860, −6.62956621211911120997267001117, −6.08797782100849683859174267433, −5.01874618387000423964225232535, −3.71320592245136289916551990840, −2.24040738187388492336580816587,
2.24040738187388492336580816587, 3.71320592245136289916551990840, 5.01874618387000423964225232535, 6.08797782100849683859174267433, 6.62956621211911120997267001117, 8.694912320601476106960796099860, 9.418352849880820091555957559356, 10.51701510715389623659238517197, 11.64390399044447017800422711859, 12.94959674760005301372362375138
