| L(s) = 1 | − 1.31·5-s + 4.77i·7-s + 6.27i·11-s + 8.24·17-s + 4.35i·19-s − 4i·23-s − 3.27·25-s − 6.27i·35-s + 7.40i·43-s − 10.2i·47-s − 15.8·49-s − 8.24i·55-s + 3.72·61-s − 16.8·73-s − 29.9·77-s + ⋯ |
| L(s) = 1 | − 0.587·5-s + 1.80i·7-s + 1.89i·11-s + 1.99·17-s + 0.999i·19-s − 0.834i·23-s − 0.654·25-s − 1.06i·35-s + 1.12i·43-s − 1.49i·47-s − 2.26·49-s − 1.11i·55-s + 0.476·61-s − 1.96·73-s − 3.41·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294784147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294784147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 4.77iT - 7T^{2} \) |
| 11 | \( 1 - 6.27iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 8.24T + 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 7.40iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.181238903540988495529332532369, −8.221202474584321106252036743098, −7.81702502100833304444104248238, −6.92794922100425259875631233019, −5.91803009161007422643688433640, −5.34398836715200928504090539134, −4.48656172447263170959769735221, −3.48329878443316619984704325861, −2.48112936985396120042962119439, −1.62496728190546380222102477356,
0.46760143025212104560629593514, 1.22086474414416192580514730087, 3.18649142576214171978862032546, 3.53209194001296771936305634742, 4.38918621096297507239320336322, 5.46256633152382359912824683433, 6.18484465429831044114832378611, 7.29479727233705336482075304768, 7.62876956029155700929469975815, 8.329931417392539585150083624957
