| L(s) = 1 | + (−0.984 − 0.173i)2-s + (−0.567 + 0.809i)3-s + (0.939 + 0.342i)4-s + (0.221 − 2.22i)5-s + (0.699 − 0.699i)6-s + (−3.29 − 0.288i)7-s + (−0.866 − 0.5i)8-s + (0.691 + 1.90i)9-s + (−0.604 + 2.15i)10-s + (−1.00 − 0.579i)11-s + (−0.809 + 0.567i)12-s + (−1.84 + 5.08i)13-s + (3.19 + 0.856i)14-s + (1.67 + 1.44i)15-s + (0.766 + 0.642i)16-s + (1.71 − 0.624i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.327 + 0.467i)3-s + (0.469 + 0.171i)4-s + (0.0989 − 0.995i)5-s + (0.285 − 0.285i)6-s + (−1.24 − 0.108i)7-s + (−0.306 − 0.176i)8-s + (0.230 + 0.633i)9-s + (−0.191 + 0.680i)10-s + (−0.302 − 0.174i)11-s + (−0.233 + 0.163i)12-s + (−0.512 + 1.40i)13-s + (0.854 + 0.228i)14-s + (0.432 + 0.372i)15-s + (0.191 + 0.160i)16-s + (0.415 − 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.102211 + 0.269917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.102211 + 0.269917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 5 | \( 1 + (-0.221 + 2.22i)T \) |
| 37 | \( 1 + (4.38 - 4.21i)T \) |
| good | 3 | \( 1 + (0.567 - 0.809i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (3.29 + 0.288i)T + (6.89 + 1.21i)T^{2} \) |
| 11 | \( 1 + (1.00 + 0.579i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.84 - 5.08i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.71 + 0.624i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (1.54 - 2.20i)T + (-6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (4.41 - 2.54i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.84 - 6.87i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (3.41 + 3.41i)T + 31iT^{2} \) |
| 41 | \( 1 + (0.737 - 2.02i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 5.55iT - 43T^{2} \) |
| 47 | \( 1 + (1.24 - 4.64i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.83 + 0.160i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (0.0217 + 0.248i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (2.24 + 4.82i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-0.538 + 6.16i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.277 - 1.57i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-8.83 - 8.83i)T + 73iT^{2} \) |
| 79 | \( 1 + (-1.01 - 0.0886i)T + (77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (-6.90 + 14.8i)T + (-53.3 - 63.5i)T^{2} \) |
| 89 | \( 1 + (-2.22 + 0.194i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (7.12 + 12.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.72914341960902030671687582779, −10.57762880595839449350863712866, −9.781274847314718191143078257697, −9.283812823133959289030323308034, −8.195910550089210989922633026218, −7.12599700300911620000992335985, −5.99794623400216191139484119765, −4.87432214954533946680480731748, −3.67956533277169857374828127121, −1.89399555154889832527047514911,
0.23699807875477819977119956079, 2.47539503565535588389014584740, 3.54866541701602471148202058222, 5.68090011564705248592904970628, 6.43792725366344268040946805199, 7.16115578057728104918905793412, 8.071215688418656724568663698557, 9.470529335464236593510836807313, 10.10029105187737678420833827872, 10.75403285325359443989685191186
