L(s) = 1 | − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s + 17-s + (0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + (−0.499 − 0.866i)25-s − 27-s − 0.999·28-s + ⋯ |
L(s) = 1 | − 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + 9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s + 17-s + (0.499 + 0.866i)20-s + (−0.5 − 0.866i)21-s + (−0.499 − 0.866i)25-s − 27-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6125779443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6125779443\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−12.03701092219879091315253732064, −11.52990567531916683707658013002, −9.809499518364558921841200811466, −9.379129554727138987262181592000, −8.230843306843809813624834981892, −7.21199101013761836299966820716, −5.88403537497648354489194189545, −4.93680935064037131939321005328, −4.17047827773770065875382802897, −1.94850255007359042452430354403,
1.30636796685490247482543477442, 3.60097272772237206383982674068, 5.05318271691171487941791541812, 5.76417212870445232069862411033, 6.73398311178070495164713305563, 7.74128898570715815818246340046, 9.330075399939161905805593651989, 10.25632170110440673772710657933, 10.71771737634813806185965207332, 11.42705320179071613570104099214