L(s) = 1 | + 1.72i·2-s − i·3-s − 0.979·4-s − 1.65i·5-s + 1.72·6-s + 1.76i·8-s − 9-s + 2.85·10-s + (−2.71 + 1.89i)11-s + 0.979i·12-s + 2.97·13-s − 1.65·15-s − 4.99·16-s − 1.49·17-s − 1.72i·18-s − 5.51·19-s + ⋯ |
L(s) = 1 | + 1.22i·2-s − 0.577i·3-s − 0.489·4-s − 0.740i·5-s + 0.704·6-s + 0.622i·8-s − 0.333·9-s + 0.903·10-s + (−0.820 + 0.572i)11-s + 0.282i·12-s + 0.825·13-s − 0.427·15-s − 1.24·16-s − 0.362·17-s − 0.406i·18-s − 1.26·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9396870384\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9396870384\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.71 - 1.89i)T \) |
good | 2 | \( 1 - 1.72iT - 2T^{2} \) |
| 5 | \( 1 + 1.65iT - 5T^{2} \) |
| 13 | \( 1 - 2.97T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 5.51T + 19T^{2} \) |
| 23 | \( 1 + 8.62T + 23T^{2} \) |
| 29 | \( 1 - 8.02iT - 29T^{2} \) |
| 31 | \( 1 - 8.86iT - 31T^{2} \) |
| 37 | \( 1 - 8.22T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 - 5.03iT - 43T^{2} \) |
| 47 | \( 1 - 7.36iT - 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 + 4.96iT - 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 - 9.85T + 73T^{2} \) |
| 79 | \( 1 - 3.93iT - 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.16iT - 89T^{2} \) |
| 97 | \( 1 + 3.35iT - 97T^{2} \) |
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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
−9.371137034630296624021074647507, −8.552998264301419195842196130771, −8.136780744591722775367308445690, −7.38211853921616663914994211635, −6.48949893962959288877842677862, −5.96537000854666038403508169571, −5.01993971664078034845972075625, −4.29311856551989919565196234287, −2.67253646774007919717206598613, −1.54183565707230993990598066961,
0.33952501662030287806166174668, 2.17667579560171458622184026137, 2.73322657566312930603938413870, 3.89976548535606301659271007774, 4.28639151955524488865641745449, 5.92109986351719385938337976464, 6.30123695416974217729109916045, 7.62383227784281909422740568788, 8.402957328362867450379635200333, 9.343194844746102259651960755782