| 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 | \( 1 + (-371. + 269. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.20454430512003904572062796918, −11.66343838159726760163804691365, −10.85513533215892925725897759744, −9.480332307336773305930941961711, −8.511943109108459532313536483283, −7.02864715151737907019530649746, −6.54712741548742782712809445395, −5.01490472347821962335836297132, −2.50533952858152833386837121662, −0.57986836463863395141316239489,
1.93452712643048490458752034847, 3.97674985518514192265760764559, 5.18979198249897867848906790337, 6.89348819102457583481031484948, 8.298563470958815006595567557311, 9.450216862940904127594602840275, 10.35703366038101939533635151140, 10.92386948301547131530902114606, 12.49281956019886379658156678095, 13.09329742092234896384964761634
