L(s) = 1 | + (−1.27 − 1.16i)3-s + (0.307 − 2.21i)5-s + (−3.28 − 1.89i)7-s + (0.264 + 2.98i)9-s + (−1.86 + 3.23i)11-s + (−1.09 + 0.634i)13-s + (−2.98 + 2.46i)15-s − 0.973i·17-s + 2.74·19-s + (1.97 + 6.26i)21-s + (−5.83 + 3.36i)23-s + (−4.81 − 1.36i)25-s + (3.15 − 4.12i)27-s + (−1.99 + 3.45i)29-s + (−5.36 − 9.28i)31-s + ⋯ |
L(s) = 1 | + (−0.737 − 0.675i)3-s + (0.137 − 0.990i)5-s + (−1.24 − 0.716i)7-s + (0.0882 + 0.996i)9-s + (−0.562 + 0.974i)11-s + (−0.304 + 0.176i)13-s + (−0.770 + 0.637i)15-s − 0.236i·17-s + 0.629·19-s + (0.431 + 1.36i)21-s + (−1.21 + 0.702i)23-s + (−0.962 − 0.272i)25-s + (0.607 − 0.794i)27-s + (−0.370 + 0.641i)29-s + (−0.962 − 1.66i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0379226 + 0.336692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0379226 + 0.336692i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.27 + 1.16i)T \) |
| 5 | \( 1 + (-0.307 + 2.21i)T \) |
good | 7 | \( 1 + (3.28 + 1.89i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.86 - 3.23i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.09 - 0.634i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.973iT - 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + (5.83 - 3.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.99 - 3.45i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.36 + 9.28i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.59iT - 37T^{2} \) |
| 41 | \( 1 + (2.92 + 5.07i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.95 - 2.86i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.18 + 2.41i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + (3.98 + 6.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.29 + 7.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.993 + 0.573i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.20T + 71T^{2} \) |
| 73 | \( 1 - 0.990iT - 73T^{2} \) |
| 79 | \( 1 + (-7.83 + 13.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.08 - 0.625i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.97T + 89T^{2} \) |
| 97 | \( 1 + (5.56 + 3.21i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.01684528772255215736771410812, −9.893440029707795092546604954043, −9.400052578319845176319433725454, −7.76801388658556609746813512284, −7.23955508524299487346221442438, −6.04233163872715351032746981915, −5.16277471498769774852212757103, −3.94725999473629772376091706467, −1.99879160310702784052968958870, −0.23748366646979550554632543074,
2.81966146481237173331387829853, 3.65324616031228845195363012868, 5.35265564827187458794087275012, 6.10357653968795985912408751401, 6.83536535355761550388432337033, 8.308581783907912737947042165565, 9.533187047172640708974322420852, 10.12562192053837299874212645078, 10.90192713795466771433135311278, 11.81495166108392686812664038389