| L(s) = 1 | − 1.43·2-s + 0.0686·4-s + 5-s − 2.74·7-s + 2.77·8-s − 1.43·10-s − 2.74·11-s − 5.14·13-s + 3.94·14-s − 4.13·16-s + 3.72·17-s − 0.404·19-s + 0.0686·20-s + 3.94·22-s + 5.45·23-s + 25-s + 7.40·26-s − 0.188·28-s + 29-s + 1.45·31-s + 0.388·32-s − 5.36·34-s − 2.74·35-s + 6.76·37-s + 0.581·38-s + 2.77·40-s − 9.78·41-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.0343·4-s + 0.447·5-s − 1.03·7-s + 0.982·8-s − 0.454·10-s − 0.827·11-s − 1.42·13-s + 1.05·14-s − 1.03·16-s + 0.904·17-s − 0.0927·19-s + 0.0153·20-s + 0.841·22-s + 1.13·23-s + 0.200·25-s + 1.45·26-s − 0.0355·28-s + 0.185·29-s + 0.261·31-s + 0.0686·32-s − 0.919·34-s − 0.463·35-s + 1.11·37-s + 0.0943·38-s + 0.439·40-s − 1.52·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6464349571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6464349571\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 + 2.74T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 - 3.72T + 17T^{2} \) |
| 19 | \( 1 + 0.404T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 31 | \( 1 - 1.45T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 + 9.78T + 41T^{2} \) |
| 43 | \( 1 - 4.43T + 43T^{2} \) |
| 47 | \( 1 + 2.60T + 47T^{2} \) |
| 53 | \( 1 - 6.43T + 53T^{2} \) |
| 59 | \( 1 - 9.91T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 1.62T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 - 7.82T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.740578729232786388127976184331, −9.037040314977176693180496465746, −8.076118133452586258150504044492, −7.36970899871218492111223073568, −6.61352613631778993665096036846, −5.41505195687278823753276083115, −4.70777471881903065136635914776, −3.25606953653899693689065639521, −2.26388666047203924476833680422, −0.67013277183930208911141896232,
0.67013277183930208911141896232, 2.26388666047203924476833680422, 3.25606953653899693689065639521, 4.70777471881903065136635914776, 5.41505195687278823753276083115, 6.61352613631778993665096036846, 7.36970899871218492111223073568, 8.076118133452586258150504044492, 9.037040314977176693180496465746, 9.740578729232786388127976184331
