| L(s) = 1 | + 1.81·2-s − 28.6·4-s + 65.9·5-s + 148.·7-s − 110.·8-s + 119.·10-s + 47.5·11-s − 72.8·13-s + 270.·14-s + 717.·16-s − 1.03e3·17-s + 1.43e3·19-s − 1.89e3·20-s + 86.4·22-s + 529·23-s + 1.22e3·25-s − 132.·26-s − 4.27e3·28-s + 3.35e3·29-s − 3.47e3·31-s + 4.83e3·32-s − 1.88e3·34-s + 9.82e3·35-s + 2.08e3·37-s + 2.59e3·38-s − 7.27e3·40-s + 1.60e4·41-s + ⋯ |
| L(s) = 1 | + 0.321·2-s − 0.896·4-s + 1.17·5-s + 1.14·7-s − 0.609·8-s + 0.379·10-s + 0.118·11-s − 0.119·13-s + 0.369·14-s + 0.701·16-s − 0.870·17-s + 0.908·19-s − 1.05·20-s + 0.0380·22-s + 0.208·23-s + 0.392·25-s − 0.0384·26-s − 1.03·28-s + 0.741·29-s − 0.648·31-s + 0.834·32-s − 0.279·34-s + 1.35·35-s + 0.250·37-s + 0.291·38-s − 0.718·40-s + 1.48·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.683535611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.683535611\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 1.81T + 32T^{2} \) |
| 5 | \( 1 - 65.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 148.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 47.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 72.8T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.43e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 3.35e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.08e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.60e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.46e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.76e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.17e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.66e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.09e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.09e5T + 8.58e9T^{2} \) |
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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
−11.56014488415496222336552256496, −10.47931472956335720120454448965, −9.423044263591636884544507858245, −8.759938128802493026410774918165, −7.52215635644780458108977957419, −5.98483052337023349095698258532, −5.17310498075494605376752307170, −4.18827778085495482231175959960, −2.43573089569574635423503726622, −1.03126033348511674179198537453,
1.03126033348511674179198537453, 2.43573089569574635423503726622, 4.18827778085495482231175959960, 5.17310498075494605376752307170, 5.98483052337023349095698258532, 7.52215635644780458108977957419, 8.759938128802493026410774918165, 9.423044263591636884544507858245, 10.47931472956335720120454448965, 11.56014488415496222336552256496
