L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (2.43 + 0.653i)5-s + (0.762 + 1.32i)7-s + (0.707 − 0.707i)8-s + 2.52·10-s + 0.293·11-s + (0.472 − 1.76i)13-s + (1.07 + 1.07i)14-s + (0.500 − 0.866i)16-s + (0.0537 + 0.200i)17-s + (−0.213 + 0.798i)19-s + (2.43 − 0.653i)20-s + (0.283 − 0.0760i)22-s + (−5.10 + 5.10i)23-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (1.09 + 0.292i)5-s + (0.288 + 0.498i)7-s + (0.249 − 0.249i)8-s + 0.798·10-s + 0.0885·11-s + (0.130 − 0.488i)13-s + (0.288 + 0.288i)14-s + (0.125 − 0.216i)16-s + (0.0130 + 0.0486i)17-s + (−0.0490 + 0.183i)19-s + (0.545 − 0.146i)20-s + (0.0604 − 0.0162i)22-s + (−1.06 + 1.06i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.73825 - 0.0427329i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.73825 - 0.0427329i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (1.03 + 5.99i)T \) |
good | 5 | \( 1 + (-2.43 - 0.653i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-0.762 - 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.293T + 11T^{2} \) |
| 13 | \( 1 + (-0.472 + 1.76i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.0537 - 0.200i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (0.213 - 0.798i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.10 - 5.10i)T - 23iT^{2} \) |
| 29 | \( 1 + (-4.72 - 4.72i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4.39 + 4.39i)T - 31iT^{2} \) |
| 41 | \( 1 + (0.398 + 0.691i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.60 + 5.60i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.508iT - 47T^{2} \) |
| 53 | \( 1 + (3.09 + 1.78i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.715 - 2.66i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.820 - 0.219i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (0.106 - 0.0617i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (12.0 - 6.95i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 2.83iT - 73T^{2} \) |
| 79 | \( 1 + (0.472 - 1.76i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-4.50 - 2.59i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (10.2 - 2.74i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.15 + 4.15i)T + 97iT^{2} \) |
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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
−10.41133019542699403306666023704, −9.928387549101410087007551611651, −8.879960270491189638222623390404, −7.84018379296176998077192832986, −6.67732406991567692871778655420, −5.81887091018177316899297599993, −5.25747029464136014382062452946, −3.91450192345939765090889080035, −2.68818514106345317193103903285, −1.69660280742526287324001808128,
1.53472770501670063893661504598, 2.76366478666720728245366436854, 4.23926480270077463339941042768, 4.94580466425950200026918856330, 6.12769952246120864087696381144, 6.61785275275997298304645749052, 7.87442573688262921528128216908, 8.721718328589700416304065855167, 9.849064628322217162407913367963, 10.41510488136709464027599622455