L(s) = 1 | + (−0.0475 − 0.998i)2-s + (−1.91 + 0.560i)3-s + (−0.995 + 0.0950i)4-s + (−0.654 − 0.755i)5-s + (0.651 + 1.88i)6-s + (0.419 + 0.216i)7-s + (0.142 + 0.989i)8-s + (2.49 − 1.60i)9-s + (−0.723 + 0.690i)10-s + (1.84 − 0.739i)12-s + (0.195 − 0.428i)14-s + (1.67 + 1.07i)15-s + (0.981 − 0.189i)16-s + (−1.71 − 2.41i)18-s + (0.723 + 0.690i)20-s + (−0.921 − 0.177i)21-s + ⋯ |
L(s) = 1 | + (−0.0475 − 0.998i)2-s + (−1.91 + 0.560i)3-s + (−0.995 + 0.0950i)4-s + (−0.654 − 0.755i)5-s + (0.651 + 1.88i)6-s + (0.419 + 0.216i)7-s + (0.142 + 0.989i)8-s + (2.49 − 1.60i)9-s + (−0.723 + 0.690i)10-s + (1.84 − 0.739i)12-s + (0.195 − 0.428i)14-s + (1.67 + 1.07i)15-s + (0.981 − 0.189i)16-s + (−1.71 − 2.41i)18-s + (0.723 + 0.690i)20-s + (−0.921 − 0.177i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2541806278\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2541806278\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0475 + 0.998i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 67 | \( 1 + (0.580 + 0.814i)T \) |
good | 3 | \( 1 + (1.91 - 0.560i)T + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.419 - 0.216i)T + (0.580 + 0.814i)T^{2} \) |
| 11 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 13 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 19 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 23 | \( 1 + (-0.235 + 0.971i)T + (-0.888 - 0.458i)T^{2} \) |
| 29 | \( 1 + (0.723 + 1.25i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.911 - 1.28i)T + (-0.327 - 0.945i)T^{2} \) |
| 43 | \( 1 + (0.481 + 1.05i)T + (-0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + (1.34 + 1.28i)T + (0.0475 + 0.998i)T^{2} \) |
| 53 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (-0.428 - 1.23i)T + (-0.786 + 0.618i)T^{2} \) |
| 71 | \( 1 + (-0.981 + 0.189i)T^{2} \) |
| 73 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 79 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 83 | \( 1 + (-1.74 + 0.336i)T + (0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.783124346006185756442967801212, −8.866838525773463639384423029010, −7.968650190805365430879321353093, −6.77858778397809325247798118876, −5.67445553454277279010523014246, −4.98722533037997977882852894998, −4.43762513081758746223731501614, −3.60666954431783903759826117191, −1.62188031639464862235474013882, −0.31677996776084031456372564143,
1.39758442547092752963513007189, 3.67613317187346973151313635028, 4.73766462738717033708671572078, 5.32829405691069593279337605096, 6.26804339215334841504357546767, 6.85498017006008314907127787015, 7.49371813514451041638580579933, 8.064326291791278055791814682334, 9.502300350769391702717897215882, 10.40094156897041469874564407489