| L(s) = 1 | + (−1.61 − 1.17i)2-s + (−1.38 + 4.24i)3-s + (1.23 + 3.80i)4-s + (4.04 − 2.93i)5-s + (7.22 − 5.25i)6-s + (−0.607 − 1.86i)7-s + (2.47 − 7.60i)8-s + (5.69 + 4.13i)9-s − 10·10-s + (−16.0 + 32.7i)11-s − 17.8·12-s + (−10.8 − 7.88i)13-s + (−1.21 + 3.73i)14-s + (6.90 + 21.2i)15-s + (−12.9 + 9.40i)16-s + (−93.2 + 67.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.265 + 0.817i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.491 − 0.357i)6-s + (−0.0327 − 0.100i)7-s + (0.109 − 0.336i)8-s + (0.210 + 0.153i)9-s − 0.316·10-s + (−0.441 + 0.897i)11-s − 0.429·12-s + (−0.231 − 0.168i)13-s + (−0.0231 + 0.0713i)14-s + (0.118 + 0.365i)15-s + (−0.202 + 0.146i)16-s + (−1.33 + 0.966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.549458 + 0.668743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.549458 + 0.668743i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 + 1.17i)T \) |
| 5 | \( 1 + (-4.04 + 2.93i)T \) |
| 11 | \( 1 + (16.0 - 32.7i)T \) |
| good | 3 | \( 1 + (1.38 - 4.24i)T + (-21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 + (0.607 + 1.86i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (10.8 + 7.88i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (93.2 - 67.7i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (44.0 - 135. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-39.9 - 122. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-11.0 - 7.99i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (118. + 365. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (55.9 - 172. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 199.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (30.9 - 95.2i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (130. + 94.4i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-144. - 444. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-579. + 420. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 844.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-168. + 122. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (51.0 + 157. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-933. - 678. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (233. - 169. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 181.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-371. - 269. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.09329742092234896384964761634, −12.49281956019886379658156678095, −10.92386948301547131530902114606, −10.35703366038101939533635151140, −9.450216862940904127594602840275, −8.298563470958815006595567557311, −6.89348819102457583481031484948, −5.18979198249897867848906790337, −3.97674985518514192265760764559, −1.93452712643048490458752034847,
0.57986836463863395141316239489, 2.50533952858152833386837121662, 5.01490472347821962335836297132, 6.54712741548742782712809445395, 7.02864715151737907019530649746, 8.511943109108459532313536483283, 9.480332307336773305930941961711, 10.85513533215892925725897759744, 11.66343838159726760163804691365, 13.20454430512003904572062796918
