| L(s) = 1 | + (3.01 − 3.47i)3-s + (7.38 + 1.06i)5-s + (35.2 − 16.0i)7-s + (8.51 + 59.2i)9-s + (−14.6 − 50.0i)11-s + (124. − 272. i)13-s + (25.9 − 22.4i)15-s + (217. + 337. i)17-s + (211. − 328. i)19-s + (50.2 − 171. i)21-s + (−104. − 518. i)23-s + (−546. − 160. i)25-s + (545. + 350. i)27-s + (989. − 635. i)29-s + (905. + 1.04e3i)31-s + ⋯ |
| L(s) = 1 | + (0.334 − 0.386i)3-s + (0.295 + 0.0424i)5-s + (0.719 − 0.328i)7-s + (0.105 + 0.730i)9-s + (−0.121 − 0.413i)11-s + (0.735 − 1.61i)13-s + (0.115 − 0.0999i)15-s + (0.751 + 1.16i)17-s + (0.585 − 0.910i)19-s + (0.113 − 0.387i)21-s + (−0.197 − 0.980i)23-s + (−0.874 − 0.256i)25-s + (0.747 + 0.480i)27-s + (1.17 − 0.756i)29-s + (0.941 + 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.06825 - 0.685663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06825 - 0.685663i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (104. + 518. i)T \) |
| good | 3 | \( 1 + (-3.01 + 3.47i)T + (-11.5 - 80.1i)T^{2} \) |
| 5 | \( 1 + (-7.38 - 1.06i)T + (599. + 176. i)T^{2} \) |
| 7 | \( 1 + (-35.2 + 16.0i)T + (1.57e3 - 1.81e3i)T^{2} \) |
| 11 | \( 1 + (14.6 + 50.0i)T + (-1.23e4 + 7.91e3i)T^{2} \) |
| 13 | \( 1 + (-124. + 272. i)T + (-1.87e4 - 2.15e4i)T^{2} \) |
| 17 | \( 1 + (-217. - 337. i)T + (-3.46e4 + 7.59e4i)T^{2} \) |
| 19 | \( 1 + (-211. + 328. i)T + (-5.41e4 - 1.18e5i)T^{2} \) |
| 29 | \( 1 + (-989. + 635. i)T + (2.93e5 - 6.43e5i)T^{2} \) |
| 31 | \( 1 + (-905. - 1.04e3i)T + (-1.31e5 + 9.14e5i)T^{2} \) |
| 37 | \( 1 + (1.08e3 - 156. i)T + (1.79e6 - 5.28e5i)T^{2} \) |
| 41 | \( 1 + (243. - 1.69e3i)T + (-2.71e6 - 7.96e5i)T^{2} \) |
| 43 | \( 1 + (292. + 253. i)T + (4.86e5 + 3.38e6i)T^{2} \) |
| 47 | \( 1 + 2.36e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (3.77e3 - 1.72e3i)T + (5.16e6 - 5.96e6i)T^{2} \) |
| 59 | \( 1 + (1.48e3 - 3.25e3i)T + (-7.93e6 - 9.15e6i)T^{2} \) |
| 61 | \( 1 + (2.72e3 - 2.36e3i)T + (1.97e6 - 1.37e7i)T^{2} \) |
| 67 | \( 1 + (-241. + 822. i)T + (-1.69e7 - 1.08e7i)T^{2} \) |
| 71 | \( 1 + (-4.67e3 - 1.37e3i)T + (2.13e7 + 1.37e7i)T^{2} \) |
| 73 | \( 1 + (4.91e3 + 3.15e3i)T + (1.17e7 + 2.58e7i)T^{2} \) |
| 79 | \( 1 + (-8.47e3 - 3.86e3i)T + (2.55e7 + 2.94e7i)T^{2} \) |
| 83 | \( 1 + (-1.07e4 + 1.53e3i)T + (4.55e7 - 1.33e7i)T^{2} \) |
| 89 | \( 1 + (-327. - 283. i)T + (8.92e6 + 6.21e7i)T^{2} \) |
| 97 | \( 1 + (1.07e4 + 1.53e3i)T + (8.49e7 + 2.49e7i)T^{2} \) |
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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
−13.43549231096318353426359437729, −12.32143596472395720390641809961, −10.85654968971399887250322353746, −10.21699321951542295569570526454, −8.306451120607532815143877227610, −7.935916569612393023459195918117, −6.22608112917320217395177519498, −4.87878209728362052697822264423, −2.98454388451741256704876949738, −1.23255067633872852366387917376,
1.64120383083100155345402937618, 3.58053773197873194789263578409, 5.00456855902977067750625951246, 6.48292423917350867471165693222, 7.931358812689828503748022984677, 9.225582214362856958559720798047, 9.869150906356023324169887096904, 11.51764133323496642588595018296, 12.11447760461834340489859730173, 13.83324366022001692996249080987
