L(s) = 1 | + (−0.842 − 1.13i)2-s + (0.218 + 1.71i)3-s + (−0.581 + 1.91i)4-s + (1.76 − 1.69i)6-s + 3.64·7-s + (2.66 − 0.949i)8-s + (−2.90 + 0.750i)9-s − 5.07i·11-s + (−3.41 − 0.581i)12-s + 1.70·13-s + (−3.06 − 4.14i)14-s + (−3.32 − 2.22i)16-s + 4.08·17-s + (3.29 + 2.66i)18-s + 1.26·19-s + ⋯ |
L(s) = 1 | + (−0.595 − 0.803i)2-s + (0.126 + 0.992i)3-s + (−0.290 + 0.956i)4-s + (0.721 − 0.691i)6-s + 1.37·7-s + (0.941 − 0.335i)8-s + (−0.968 + 0.250i)9-s − 1.52i·11-s + (−0.985 − 0.168i)12-s + 0.473·13-s + (−0.820 − 1.10i)14-s + (−0.830 − 0.556i)16-s + 0.989·17-s + (0.777 + 0.628i)18-s + 0.290·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30090 + 0.00345145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30090 + 0.00345145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.842 + 1.13i)T \) |
| 3 | \( 1 + (-0.218 - 1.71i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.07iT - 11T^{2} \) |
| 13 | \( 1 - 1.70T + 13T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 4.70iT - 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 - 4.86iT - 31T^{2} \) |
| 37 | \( 1 - 7.56T + 37T^{2} \) |
| 41 | \( 1 - 1.50iT - 41T^{2} \) |
| 43 | \( 1 + 3.43iT - 43T^{2} \) |
| 47 | \( 1 - 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.87iT - 53T^{2} \) |
| 59 | \( 1 + 0.788iT - 59T^{2} \) |
| 61 | \( 1 - 0.627iT - 61T^{2} \) |
| 67 | \( 1 - 4.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.21T + 71T^{2} \) |
| 73 | \( 1 - 4.21iT - 73T^{2} \) |
| 79 | \( 1 - 0.992iT - 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 + 11.5iT - 89T^{2} \) |
| 97 | \( 1 + 7.40iT - 97T^{2} \) |
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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
−10.74562688040124088196985667401, −9.899027588086896850478905477310, −8.936138755190377799469027402902, −8.308415753534756896409817923423, −7.67233181481498524362290839282, −5.82769372089458168696275289743, −4.89641648883402281636130180797, −3.78018377629706964086104106646, −2.93664428919771614263526792890, −1.24399258855644690271547033737,
1.18677140194481391223941513813, 2.21838706209154803012952388394, 4.40699524447601469099509021843, 5.36240260232390987132240616886, 6.35910835693131689606536435685, 7.39294504437022565042275184026, 7.82812253052311253019126451722, 8.597995576123720226715721480156, 9.592573341267446919573186596709, 10.57264904579678912798364424697