| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 2i·7-s + i·8-s + 2·9-s − 3·11-s + i·12-s + 4i·13-s + 2·14-s + 16-s − 3i·17-s − 2i·18-s − 5·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353i·8-s + 0.666·9-s − 0.904·11-s + 0.288i·12-s + 1.10i·13-s + 0.534·14-s + 0.250·16-s − 0.727i·17-s − 0.471i·18-s − 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679488 - 0.419947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679488 - 0.419947i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 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
−15.32586913450335375715752042507, −13.97638971110956153437607603611, −12.86398378257962693813870112174, −12.11356300791817569216803204549, −10.81916518970544507153846230906, −9.490929936023550614593847231485, −8.178633051113435903472474160629, −6.55588619229359896656962840043, −4.64766923819025120481399176619, −2.31774854305621554289445648670,
3.93792105454979089855934684109, 5.42915259659245748724299711985, 7.15664705345784741772316209225, 8.328131252201877576778919289006, 9.993592698117683599960581114453, 10.68422598307452074816936384501, 12.73912890563041417024438280589, 13.55268857495918184104546717963, 15.09302265541016053388976990517, 15.59163635768525087204604244357
