| L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s + 5-s + 0.470·6-s + 2.52·7-s − 1.77·8-s + 9-s + 0.470·10-s + 1.05·11-s − 1.77·12-s + 13-s + 1.19·14-s + 15-s + 2.71·16-s + 2·17-s + 0.470·18-s + 0.692·19-s − 1.77·20-s + 2.52·21-s + 0.498·22-s + 2.94·23-s − 1.77·24-s + 25-s + 0.470·26-s + 27-s − 4.49·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s + 0.447·5-s + 0.192·6-s + 0.955·7-s − 0.628·8-s + 0.333·9-s + 0.148·10-s + 0.319·11-s − 0.513·12-s + 0.277·13-s + 0.318·14-s + 0.258·15-s + 0.679·16-s + 0.485·17-s + 0.110·18-s + 0.158·19-s − 0.397·20-s + 0.551·21-s + 0.106·22-s + 0.613·23-s − 0.363·24-s + 0.200·25-s + 0.0923·26-s + 0.192·27-s − 0.850·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.201441151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.201441151\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.692T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 - 6.52T + 43T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + 9.80T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 8.08T + 67T^{2} \) |
| 71 | \( 1 - 2.11T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 6.16T + 83T^{2} \) |
| 89 | \( 1 - 6.49T + 89T^{2} \) |
| 97 | \( 1 + 2.64T + 97T^{2} \) |
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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
−8.101113730328658039896998794824, −7.62524514876116570447771604813, −6.53294909004782982352808766639, −5.87037978545541593957829147185, −4.88190214076937355126681770423, −4.67150566428154809560139785982, −3.61002849570542895341147920731, −2.97928285030383645024387718130, −1.81121040534846661948298157655, −0.939702153000281776438282092936,
0.939702153000281776438282092936, 1.81121040534846661948298157655, 2.97928285030383645024387718130, 3.61002849570542895341147920731, 4.67150566428154809560139785982, 4.88190214076937355126681770423, 5.87037978545541593957829147185, 6.53294909004782982352808766639, 7.62524514876116570447771604813, 8.101113730328658039896998794824
