| L(s) = 1 | + (−0.451 + 1.34i)2-s + (2.75 + 1.19i)3-s + (−1.59 − 1.21i)4-s + (7.91 + 3.15i)5-s + (−2.84 + 3.15i)6-s + (−1.21 + 0.562i)7-s + (2.34 − 1.58i)8-s + (6.15 + 6.56i)9-s + (−7.79 + 9.17i)10-s + (−7.84 − 2.17i)11-s + (−2.93 − 5.23i)12-s + (−11.2 + 21.2i)13-s + (−0.204 − 1.88i)14-s + (18.0 + 18.1i)15-s + (1.07 + 3.85i)16-s + (10.7 − 23.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.225 + 0.670i)2-s + (0.917 + 0.397i)3-s + (−0.398 − 0.302i)4-s + (1.58 + 0.630i)5-s + (−0.473 + 0.525i)6-s + (−0.173 + 0.0804i)7-s + (0.292 − 0.198i)8-s + (0.683 + 0.729i)9-s + (−0.779 + 0.917i)10-s + (−0.712 − 0.197i)11-s + (−0.244 − 0.435i)12-s + (−0.866 + 1.63i)13-s + (−0.0146 − 0.134i)14-s + (1.20 + 1.20i)15-s + (0.0668 + 0.240i)16-s + (0.634 − 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.47773 + 1.88665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47773 + 1.88665i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.451 - 1.34i)T \) |
| 3 | \( 1 + (-2.75 - 1.19i)T \) |
| 59 | \( 1 + (-28.2 + 51.7i)T \) |
| good | 5 | \( 1 + (-7.91 - 3.15i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (1.21 - 0.562i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (7.84 + 2.17i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (11.2 - 21.2i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (-10.7 + 23.3i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-19.4 + 4.28i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (-1.45 - 0.238i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (-1.33 - 3.96i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (39.1 + 8.60i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-2.99 + 4.41i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-61.6 + 10.1i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (12.3 + 44.4i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (13.4 - 5.36i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (26.9 - 22.9i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (38.9 + 13.1i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-7.84 - 11.5i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (48.5 - 19.3i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-60.2 + 6.55i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-35.8 + 21.5i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (33.1 - 1.79i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (33.8 + 100. i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-46.1 - 5.01i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−11.22613012079162073096177617232, −10.11113378277033524983913872827, −9.435864433288642861491181717698, −9.159392950954617918403157359896, −7.53814565062148052078251058355, −6.97699896387279340456177895151, −5.67252607731472346666017744451, −4.79486143425899356350983664241, −3.02799790142891571437329143511, −1.99424210088039560972260922455,
1.16773703405595611113616852494, 2.30942518132245860599792091868, 3.31856337227092798198039649391, 5.06338249007659439154965720459, 5.96003879625783021074651493793, 7.56963819273726776799022728757, 8.267198513957463763648728131228, 9.403981076528454982177925258981, 9.926103499742738397386925155129, 10.56017182030258339141114921001
