L(s) = 1 | + (−0.965 + 0.258i)2-s + (1.73 + 0.0795i)3-s + (0.866 − 0.499i)4-s + (−0.313 − 0.313i)5-s + (−1.69 + 0.370i)6-s + (−0.0745 + 0.278i)7-s + (−0.707 + 0.707i)8-s + (2.98 + 0.275i)9-s + (0.383 + 0.221i)10-s + (−0.150 − 0.563i)11-s + (1.53 − 0.796i)12-s + (−1.79 + 3.12i)13-s − 0.288i·14-s + (−0.517 − 0.567i)15-s + (0.500 − 0.866i)16-s + (−2.79 − 4.84i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.998 + 0.0459i)3-s + (0.433 − 0.249i)4-s + (−0.140 − 0.140i)5-s + (−0.690 + 0.151i)6-s + (−0.0281 + 0.105i)7-s + (−0.249 + 0.249i)8-s + (0.995 + 0.0917i)9-s + (0.121 + 0.0700i)10-s + (−0.0454 − 0.169i)11-s + (0.444 − 0.229i)12-s + (−0.496 + 0.867i)13-s − 0.0770i·14-s + (−0.133 − 0.146i)15-s + (0.125 − 0.216i)16-s + (−0.678 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.897149 + 0.0813565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.897149 + 0.0813565i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 - 0.0795i)T \) |
| 13 | \( 1 + (1.79 - 3.12i)T \) |
good | 5 | \( 1 + (0.313 + 0.313i)T + 5iT^{2} \) |
| 7 | \( 1 + (0.0745 - 0.278i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.150 + 0.563i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.79 + 4.84i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.79 + 1.81i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (3.32 - 5.76i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.57 - 2.06i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.03 - 1.03i)T - 31iT^{2} \) |
| 37 | \( 1 + (-6.72 + 1.80i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.36 + 1.97i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (3.26 - 1.88i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.71 - 3.71i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.64iT - 53T^{2} \) |
| 59 | \( 1 + (-3.29 - 0.881i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (5.25 + 9.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 - 8.55i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.98 + 14.8i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-3.52 - 3.52i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + (-8.23 - 8.23i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.64 + 13.6i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.59 - 0.694i)T + (84.0 + 48.5i)T^{2} \) |
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
−14.55299505727916212341205562331, −13.64708313608005216943725208744, −12.32927337919761791092082365974, −11.00719039803608678476597237266, −9.646254299324990271718437160140, −8.895108233900281771634528627451, −7.77532382286985354468170950158, −6.60858545746184867538943732661, −4.44211712062665931512074477069, −2.39449528467691720530944554359,
2.34275675523666473044846469699, 4.07641749843765235876016347203, 6.49996029464385801436391250431, 7.88656003652607058517979901626, 8.613053818301682148318463255976, 9.965867861437631100765269549410, 10.75266794244188091642774955194, 12.43989411568034970309111482560, 13.18706776357299593689498090174, 14.77228165557714755869520130542