L(s) = 1 | + (−0.888 + 0.458i)3-s + (0.928 − 0.371i)4-s + (−0.0947 − 1.98i)7-s + (0.580 − 0.814i)9-s + (−0.654 + 0.755i)12-s + (−1.74 + 0.336i)13-s + (0.723 − 0.690i)16-s + (0.327 − 0.566i)19-s + (0.995 + 1.72i)21-s + (0.841 + 0.540i)25-s + (−0.142 + 0.989i)27-s + (−0.827 − 1.81i)28-s + (0.888 + 0.458i)31-s + (0.235 − 0.971i)36-s + (0.959 + 1.66i)37-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.458i)3-s + (0.928 − 0.371i)4-s + (−0.0947 − 1.98i)7-s + (0.580 − 0.814i)9-s + (−0.654 + 0.755i)12-s + (−1.74 + 0.336i)13-s + (0.723 − 0.690i)16-s + (0.327 − 0.566i)19-s + (0.995 + 1.72i)21-s + (0.841 + 0.540i)25-s + (−0.142 + 0.989i)27-s + (−0.827 − 1.81i)28-s + (0.888 + 0.458i)31-s + (0.235 − 0.971i)36-s + (0.959 + 1.66i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 597 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8070439730\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8070439730\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.888 - 0.458i)T \) |
| 199 | \( 1 + (0.654 - 0.755i)T \) |
good | 2 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 5 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (0.0947 + 1.98i)T + (-0.995 + 0.0950i)T^{2} \) |
| 11 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 13 | \( 1 + (1.74 - 0.336i)T + (0.928 - 0.371i)T^{2} \) |
| 17 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 29 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 31 | \( 1 + (-0.888 - 0.458i)T + (0.580 + 0.814i)T^{2} \) |
| 37 | \( 1 + (-0.959 - 1.66i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 43 | \( 1 + (-0.142 + 0.246i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 53 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (1.38 - 0.407i)T + (0.841 - 0.540i)T^{2} \) |
| 67 | \( 1 + (0.0913 + 0.0268i)T + (0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 73 | \( 1 + (-1.07 - 0.431i)T + (0.723 + 0.690i)T^{2} \) |
| 79 | \( 1 + (0.947 - 0.903i)T + (0.0475 - 0.998i)T^{2} \) |
| 83 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 89 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 97 | \( 1 + (-1.39 - 0.720i)T + (0.580 + 0.814i)T^{2} \) |
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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
−10.71131490510391209693833780318, −10.10440267443802579023566092999, −9.555904704425739385316735791135, −7.68563459489845081235936345751, −7.00343184726242692235596048701, −6.51619142277267723439479490668, −5.07760238527539239890146644881, −4.42229286403725429182924436682, −3.03442322896134798156811609965, −1.10813912045858167593571129794,
2.11701403307844346826927620787, 2.79990032064270687708769936711, 4.81230659152222745443354638028, 5.72722420219007076273205120450, 6.33944896532815569533964025198, 7.41977885444037474624083556331, 8.108786044600909561426681095325, 9.327412595808767755244757478517, 10.26415766709156158128342327645, 11.27252663862659046256780029360