L(s) = 1 | + (0.952 + 1.65i)2-s + (−0.214 − 0.371i)3-s + (−0.815 + 1.41i)4-s − 1.47·5-s + (0.408 − 0.707i)6-s + 0.702·8-s + (1.40 − 2.43i)9-s + (−1.40 − 2.43i)10-s + (2.19 + 3.80i)11-s + 0.698·12-s + (2.69 + 2.39i)13-s + (0.315 + 0.546i)15-s + (2.30 + 3.98i)16-s + (0.601 − 1.04i)17-s + 5.36·18-s + (−1.62 + 2.80i)19-s + ⋯ |
L(s) = 1 | + (0.673 + 1.16i)2-s + (−0.123 − 0.214i)3-s + (−0.407 + 0.706i)4-s − 0.658·5-s + (0.166 − 0.288i)6-s + 0.248·8-s + (0.469 − 0.813i)9-s + (−0.443 − 0.768i)10-s + (0.662 + 1.14i)11-s + 0.201·12-s + (0.748 + 0.663i)13-s + (0.0814 + 0.141i)15-s + (0.575 + 0.996i)16-s + (0.145 − 0.252i)17-s + 1.26·18-s + (−0.371 + 0.644i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0662 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40909 + 1.50575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40909 + 1.50575i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.69 - 2.39i)T \) |
good | 2 | \( 1 + (-0.952 - 1.65i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (0.214 + 0.371i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.601 + 1.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.62 - 2.80i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.21 - 3.84i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.24T + 31T^{2} \) |
| 37 | \( 1 + (3.52 + 6.10i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 0.141T + 53T^{2} \) |
| 59 | \( 1 + (-2.67 + 4.62i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.77 - 9.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.06 + 3.58i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 0.774T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + (3.27 + 5.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.90702266304365590325204561056, −9.787824294091729799243845524717, −8.887781003680097860608983447907, −7.75982460978844246548942432655, −7.07303083264631803392344912710, −6.48032133730041205008355964857, −5.50844250991092257129829268789, −4.21920701979160949492752006898, −3.84455801116207752290261234343, −1.57132682732571764607067727754,
1.11186390317145017501669058865, 2.68869827197297701579190980266, 3.68811983037116260874689024031, 4.41789106986790159110214975531, 5.44531496134691402033899011491, 6.66472314872744668763176406787, 7.921916494084581499168982952774, 8.577847487913876485530147601375, 9.905589574125748778317049366988, 10.78163355720228341724610990706