| L(s) = 1 | + 0.792·2-s − 1.37·4-s − 3.37·5-s − 2.52·7-s − 2.67·8-s − 2.67·10-s + 5.84·13-s − 2·14-s + 0.627·16-s + 2.67·17-s + 0.939·19-s + 4.62·20-s − 2·23-s + 6.37·25-s + 4.62·26-s + 3.46·28-s − 0.792·29-s + 1.62·31-s + 5.84·32-s + 2.11·34-s + 8.51·35-s + 5·37-s + 0.744·38-s + 9.01·40-s − 10.8·41-s + 6.63·43-s − 1.58·46-s + ⋯ |
| L(s) = 1 | + 0.560·2-s − 0.686·4-s − 1.50·5-s − 0.954·7-s − 0.944·8-s − 0.844·10-s + 1.61·13-s − 0.534·14-s + 0.156·16-s + 0.648·17-s + 0.215·19-s + 1.03·20-s − 0.417·23-s + 1.27·25-s + 0.907·26-s + 0.654·28-s − 0.147·29-s + 0.292·31-s + 1.03·32-s + 0.363·34-s + 1.43·35-s + 0.821·37-s + 0.120·38-s + 1.42·40-s − 1.70·41-s + 1.01·43-s − 0.233·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030961196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030961196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 5 | \( 1 + 3.37T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 - 0.939T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 0.792T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 5.98T + 61T^{2} \) |
| 67 | \( 1 + 1.11T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.15T + 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 - 0.627T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−9.831328178356190437070853201097, −8.880851184601707278697834858680, −8.305548445921314219213501365251, −7.42767989539387870356800633748, −6.34370054895755568446349096744, −5.58536409840348011272577857205, −4.33202545606852529410059155138, −3.72211247330740265275086492277, −3.12387589977623021308529661498, −0.71270482416430164752155725411,
0.71270482416430164752155725411, 3.12387589977623021308529661498, 3.72211247330740265275086492277, 4.33202545606852529410059155138, 5.58536409840348011272577857205, 6.34370054895755568446349096744, 7.42767989539387870356800633748, 8.305548445921314219213501365251, 8.880851184601707278697834858680, 9.831328178356190437070853201097
