L(s) = 1 | + i·2-s + 3i·3-s − 4-s − 3·6-s − 5i·7-s − i·8-s − 6·9-s − 4·11-s − 3i·12-s + i·13-s + 5·14-s + 16-s − 3i·17-s − 6i·18-s − 19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 1.88i·7-s − 0.353i·8-s − 2·9-s − 1.20·11-s − 0.866i·12-s + 0.277i·13-s + 1.33·14-s + 0.250·16-s − 0.727i·17-s − 1.41i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.556556 - 0.131385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.556556 - 0.131385i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 23 | \( 1 + 7iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 13iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 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
−10.13005754909113889890701048623, −9.244549850815312309721075645447, −8.340239925267233839920725068411, −7.48816219465705187636854610825, −6.59378533420009187135711703750, −5.33685527201188894107005962850, −4.59185789638246659387353467167, −4.06614574231571688955308805184, −2.98335211476737464017735058856, −0.25849936757337310997818913150,
1.60717875927706914596224029356, 2.39703959808082716387771161589, 3.17293555578459101267045758585, 5.21063747931156207296261021631, 5.72012460110250655705893751189, 6.64126552377188817121822222429, 7.900926595129397868586337442323, 8.267657670479892378188304515591, 9.093704631429224472444185127512, 10.15016545704303514162933428331