L(s) = 1 | + (−0.527 + 0.304i)2-s + (−1.60 + 0.654i)3-s + (−0.814 + 1.41i)4-s + (0.866 + 0.5i)5-s + (0.645 − 0.833i)6-s + (1.11 − 0.646i)7-s − 2.20i·8-s + (2.14 − 2.10i)9-s − 0.608·10-s + (2.32 − 1.34i)11-s + (0.382 − 2.79i)12-s + (2.47 − 2.62i)13-s + (−0.393 + 0.681i)14-s + (−1.71 − 0.234i)15-s + (−0.957 − 1.65i)16-s − 1.63·17-s + ⋯ |
L(s) = 1 | + (−0.372 + 0.215i)2-s + (−0.925 + 0.378i)3-s + (−0.407 + 0.705i)4-s + (0.387 + 0.223i)5-s + (0.263 − 0.340i)6-s + (0.423 − 0.244i)7-s − 0.780i·8-s + (0.714 − 0.700i)9-s − 0.192·10-s + (0.700 − 0.404i)11-s + (0.110 − 0.807i)12-s + (0.686 − 0.726i)13-s + (−0.105 + 0.182i)14-s + (−0.443 − 0.0605i)15-s + (−0.239 − 0.414i)16-s − 0.396·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.891395 + 0.333824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.891395 + 0.333824i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.60 - 0.654i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-2.47 + 2.62i)T \) |
good | 2 | \( 1 + (0.527 - 0.304i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-1.11 + 0.646i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.32 + 1.34i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 - 2.80iT - 19T^{2} \) |
| 23 | \( 1 + (-4.00 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.93 - 3.35i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.48 - 4.31i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 + (7.08 + 4.09i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.05 + 10.4i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.5 + 6.07i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3.10T + 53T^{2} \) |
| 59 | \( 1 + (-8.90 - 5.14i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.42 - 4.20i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.3 + 6.57i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.938iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + (-3.27 - 5.67i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.19 + 5.30i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 + (-9.03 + 5.21i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.47192042098995015276331427915, −10.24740003029942716410824128769, −8.825452571713737206828393331859, −8.447180263344594745191694149876, −7.01125877820351459677633435769, −6.45641079255300809609432457339, −5.24030110438166011980747460792, −4.26097324394129962227800088419, −3.23568293310638052737673929458, −0.977916120882256849868789322782,
1.06755091142168028157092568740, 2.03516360025237328701692768970, 4.32202846325502867149659083747, 5.09524589590095972373369591587, 6.04380389208705520948020761571, 6.77464951313292672837277061218, 8.048554257045450977273645150333, 9.137032030150852267489387739054, 9.650905138295010618081849981037, 10.71135058528840557252446109229