| L(s) = 1 | + (−0.592 + 0.430i)2-s + (1.83 + 5.63i)3-s + (−2.30 + 7.09i)4-s + (10.4 + 7.55i)5-s + (−3.51 − 2.55i)6-s + (−5.23 + 16.0i)7-s + (−3.49 − 10.7i)8-s + (−6.58 + 4.78i)9-s − 9.41·10-s − 44.2·12-s + (60.3 − 43.8i)13-s + (−3.82 − 11.7i)14-s + (−23.5 + 72.4i)15-s + (−41.6 − 30.2i)16-s + (−66.9 − 48.6i)17-s + (1.84 − 5.66i)18-s + ⋯ |
| L(s) = 1 | + (−0.209 + 0.152i)2-s + (0.352 + 1.08i)3-s + (−0.288 + 0.887i)4-s + (0.930 + 0.675i)5-s + (−0.238 − 0.173i)6-s + (−0.282 + 0.869i)7-s + (−0.154 − 0.475i)8-s + (−0.244 + 0.177i)9-s − 0.297·10-s − 1.06·12-s + (1.28 − 0.936i)13-s + (−0.0731 − 0.224i)14-s + (−0.405 + 1.24i)15-s + (−0.650 − 0.472i)16-s + (−0.955 − 0.694i)17-s + (0.0241 − 0.0742i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.483353 + 1.60699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483353 + 1.60699i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (0.592 - 0.430i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.83 - 5.63i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-10.4 - 7.55i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (5.23 - 16.0i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-60.3 + 43.8i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (66.9 + 48.6i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-20.9 - 64.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + (52.2 - 160. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-52.9 + 38.4i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-12.6 + 38.8i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (84.9 + 261. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-22.2 - 68.3i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-120. + 87.5i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-168. + 518. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-81.9 - 59.5i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-380. - 276. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (188. - 580. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (791. - 574. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-21.1 - 15.3i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (685. - 498. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.45643102500135804239462987181, −12.52060555195562115226039953699, −11.08486631489157650468041850409, −10.03269454334212561663037635800, −9.178576078287646733775331455266, −8.412448038661552997443215413524, −6.76913959756156312412051395360, −5.47732434910503826566500086936, −3.79182394136474110386627490525, −2.75605264707211758283636070842,
0.999611715066848980111535947281, 1.94489691779320467634250774902, 4.43728675649401937206551137323, 6.03035228966966989689283990652, 6.85468317717693168932049733576, 8.460140286357567836371956304483, 9.284167785916455235313035355700, 10.36277917863077648376552686221, 11.45303829981682244267681486963, 13.20520410898496871164809107532
