L(s) = 1 | + 0.741·2-s − 1.44·4-s + (2.05 − 0.880i)5-s + (1.24 − 2.33i)7-s − 2.55·8-s + (1.52 − 0.653i)10-s − 1.41i·11-s + 5.54·13-s + (0.923 − 1.73i)14-s + 1.00·16-s − 6.07i·17-s + 7.12i·19-s + (−2.97 + 1.27i)20-s − 1.04i·22-s − 4.78·23-s + ⋯ |
L(s) = 1 | + 0.524·2-s − 0.724·4-s + (0.919 − 0.393i)5-s + (0.470 − 0.882i)7-s − 0.904·8-s + (0.482 − 0.206i)10-s − 0.426i·11-s + 1.53·13-s + (0.246 − 0.462i)14-s + 0.250·16-s − 1.47i·17-s + 1.63i·19-s + (−0.666 + 0.285i)20-s − 0.223i·22-s − 0.997·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57962 - 0.577299i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57962 - 0.577299i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2.05 + 0.880i)T \) |
| 7 | \( 1 + (-1.24 + 2.33i)T \) |
good | 2 | \( 1 - 0.741T + 2T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 + 6.07iT - 17T^{2} \) |
| 19 | \( 1 - 7.12iT - 19T^{2} \) |
| 23 | \( 1 + 4.78T + 23T^{2} \) |
| 29 | \( 1 - 5.51iT - 29T^{2} \) |
| 31 | \( 1 + 1.30iT - 31T^{2} \) |
| 37 | \( 1 + 2.57iT - 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 6.76iT - 43T^{2} \) |
| 47 | \( 1 - 7.83iT - 47T^{2} \) |
| 53 | \( 1 + 9.90T + 53T^{2} \) |
| 59 | \( 1 - 1.84T + 59T^{2} \) |
| 61 | \( 1 - 11.6iT - 61T^{2} \) |
| 67 | \( 1 - 7.23iT - 67T^{2} \) |
| 71 | \( 1 + 8.34iT - 71T^{2} \) |
| 73 | \( 1 + 0.559T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7.83iT - 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−11.65603570527138339753449952243, −10.55936596137803960080057650487, −9.695029590050996432701748431259, −8.763212954269860898239329753058, −7.900055885056632642157927458183, −6.31683476504117714991236112777, −5.52326639083389727936924168575, −4.45644880278860808691502779880, −3.39417595980536809425675077108, −1.27326428820631788152790280374,
1.97577303415852448482727393191, 3.49814552469957569328641007677, 4.78963796219847286068050057723, 5.81882062825700531438350932230, 6.46085316132515630961809408387, 8.275721737110020531407246464527, 8.887153469485413011993664962231, 9.854027339552073573701658858779, 10.86771892378664528092572353788, 11.87217323093133666325275352496