L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.909 − 1.47i)3-s + (−0.499 − 0.866i)4-s + (−3.26 − 1.88i)5-s + (0.821 + 1.52i)6-s + (−0.385 − 2.61i)7-s + 0.999·8-s + (−1.34 − 2.68i)9-s + (3.26 − 1.88i)10-s + (1.89 + 3.27i)11-s + (−1.73 − 0.0505i)12-s + (−2.90 + 2.13i)13-s + (2.45 + 0.975i)14-s + (−5.74 + 3.09i)15-s + (−0.5 + 0.866i)16-s + (1.16 + 2.00i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.525 − 0.851i)3-s + (−0.249 − 0.433i)4-s + (−1.45 − 0.842i)5-s + (0.335 + 0.622i)6-s + (−0.145 − 0.989i)7-s + 0.353·8-s + (−0.448 − 0.893i)9-s + (1.03 − 0.595i)10-s + (0.570 + 0.988i)11-s + (−0.499 − 0.0146i)12-s + (−0.805 + 0.593i)13-s + (0.657 + 0.260i)14-s + (−1.48 + 0.799i)15-s + (−0.125 + 0.216i)16-s + (0.281 + 0.487i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0186262 - 0.366691i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0186262 - 0.366691i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.909 + 1.47i)T \) |
| 7 | \( 1 + (0.385 + 2.61i)T \) |
| 13 | \( 1 + (2.90 - 2.13i)T \) |
good | 5 | \( 1 + (3.26 + 1.88i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.89 - 3.27i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 2.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.00 - 5.21i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.79 + 3.34i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.07iT - 29T^{2} \) |
| 31 | \( 1 + (4.94 + 8.56i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.92 - 3.42i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.69iT - 41T^{2} \) |
| 43 | \( 1 + 3.47T + 43T^{2} \) |
| 47 | \( 1 + (-6.71 - 3.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.653 + 0.377i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.68 + 2.12i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.37 + 4.83i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.51 + 2.60i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + (2.53 + 4.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.402 - 0.696i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.88iT - 83T^{2} \) |
| 89 | \( 1 + (13.5 + 7.80i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
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
−10.09511293503393896133307582909, −9.281741875461494750053046645552, −8.209730531938177245484169576694, −7.74354385665849718063474674594, −7.12015588219812824971929378883, −6.09790520146552711839190827995, −4.30184133465466351554675993213, −3.99591807500209348263760772744, −1.76975861119330117484996861232, −0.21954136016791554360630885845,
2.66543820275496455928150905218, 3.26594489256027391324293955607, 4.22460233403249644127816709357, 5.46096864310323604969293646878, 7.00315618213965844253141579800, 7.985715110453338675029521527094, 8.666359313876220921772085862745, 9.435996660520665465009514128811, 10.46939012270337757858813947490, 11.20033068470914397684019195564