| L(s) = 1 | + (−1.97 − 0.718i)2-s + (−2.82 + 0.997i)3-s + (0.318 + 0.267i)4-s + (4.97 − 0.522i)5-s + (6.30 + 0.0636i)6-s + (1.43 + 1.71i)7-s + (3.76 + 6.52i)8-s + (7.00 − 5.64i)9-s + (−10.1 − 2.54i)10-s + (−17.6 + 3.10i)11-s + (−1.16 − 0.438i)12-s + (−8.43 − 23.1i)13-s + (−1.60 − 4.41i)14-s + (−13.5 + 6.43i)15-s + (−3.03 − 17.2i)16-s + (0.553 − 0.958i)17-s + ⋯ |
| L(s) = 1 | + (−0.987 − 0.359i)2-s + (−0.943 + 0.332i)3-s + (0.0796 + 0.0668i)4-s + (0.994 − 0.104i)5-s + (1.05 + 0.0106i)6-s + (0.205 + 0.244i)7-s + (0.470 + 0.815i)8-s + (0.778 − 0.627i)9-s + (−1.01 − 0.254i)10-s + (−1.60 + 0.282i)11-s + (−0.0973 − 0.0365i)12-s + (−0.648 − 1.78i)13-s + (−0.114 − 0.315i)14-s + (−0.903 + 0.429i)15-s + (−0.189 − 1.07i)16-s + (0.0325 − 0.0563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.104441 - 0.306855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.104441 - 0.306855i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.82 - 0.997i)T \) |
| 5 | \( 1 + (-4.97 + 0.522i)T \) |
| good | 2 | \( 1 + (1.97 + 0.718i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 1.71i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (17.6 - 3.10i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (8.43 + 23.1i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-0.553 + 0.958i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (9.27 + 16.0i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.30 + 6.96i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (9.73 - 26.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (33.1 + 27.7i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-3.59 - 2.07i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.2 + 39.0i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (18.9 - 3.34i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-59.5 + 49.9i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 57.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-18.1 - 3.19i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-27.6 + 23.2i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-19.2 - 52.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-59.5 - 34.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (78.6 - 45.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-57.2 - 20.8i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (80.5 + 29.3i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (73.8 - 42.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-23.6 + 4.16i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.59475619610014306177786378962, −11.04461873538333964845002318963, −10.36449362414843416318401981339, −9.858135484373546110791999959755, −8.618174271273125225404913924757, −7.35701889764284517600928921649, −5.48207018492443483239616123266, −5.15119576607352756499841270661, −2.34387707921923125570451036851, −0.32576927293100731730516039672,
1.81137304161403493927144059347, 4.59895383891794123854709867687, 5.93231265478337898185002989434, 7.04107130682826426041625056339, 7.968309397557351759279784071552, 9.371961736532192595175416029278, 10.20628544519319514465341012062, 11.01788571740306638601989847669, 12.44430436924274232989526947159, 13.30325801066564050622645388487
