L(s) = 1 | + i·3-s − 9-s + 4·11-s + 2i·13-s + 2i·17-s + 4·19-s + 8i·23-s − i·27-s − 6·29-s + 8·31-s + 4i·33-s + 6i·37-s − 2·39-s − 6·41-s − 4i·43-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + 0.485i·17-s + 0.917·19-s + 1.66i·23-s − 0.192i·27-s − 1.11·29-s + 1.43·31-s + 0.696i·33-s + 0.986i·37-s − 0.320·39-s − 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.28259 + 0.792687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.28259 + 0.792687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.84689869069326518718596802165, −9.722755076682735790108006807249, −9.319980256554947494896591516093, −8.305801328331706763494157577162, −7.22689213984221627371390229595, −6.25742097909589294180313204260, −5.25755492369751829160969529345, −4.13023788952651458450579076642, −3.29839074888651189590309568332, −1.55789354773542048252640331200,
0.959303705450604034011833009324, 2.50167583361520930081768631471, 3.75594847300271695208947856097, 5.00001355591556398908592636892, 6.12870744714571890620967147062, 6.89625889003198805119507742819, 7.81492004770371017925162683408, 8.744876854152440905642778675731, 9.555110999509083129577208498245, 10.55040768262254683950084753932