| L(s) = 1 | + (1.83 − 3.55i)2-s + (−6.54 − 6.54i)3-s + (−9.26 − 13.0i)4-s + (−20.3 + 14.5i)5-s + (−35.2 + 11.2i)6-s + (24.6 + 24.6i)7-s + (−63.3 + 9.00i)8-s + 4.60i·9-s + (14.5 + 98.9i)10-s − 126. i·11-s + (−24.7 + 145. i)12-s + (−156. − 156. i)13-s + (132. − 42.4i)14-s + (228. + 37.5i)15-s + (−84.2 + 241. i)16-s + (−176. − 176. i)17-s + ⋯ |
| L(s) = 1 | + (0.458 − 0.888i)2-s + (−0.726 − 0.726i)3-s + (−0.579 − 0.815i)4-s + (−0.812 + 0.582i)5-s + (−0.979 + 0.312i)6-s + (0.503 + 0.503i)7-s + (−0.990 + 0.140i)8-s + 0.0568i·9-s + (0.145 + 0.989i)10-s − 1.04i·11-s + (−0.171 + 1.01i)12-s + (−0.923 − 0.923i)13-s + (0.678 − 0.216i)14-s + (1.01 + 0.166i)15-s + (−0.329 + 0.944i)16-s + (−0.610 − 0.610i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.103075 + 0.807748i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103075 + 0.807748i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.83 + 3.55i)T \) |
| 5 | \( 1 + (20.3 - 14.5i)T \) |
| good | 3 | \( 1 + (6.54 + 6.54i)T + 81iT^{2} \) |
| 7 | \( 1 + (-24.6 - 24.6i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 126. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (156. + 156. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (176. + 176. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 168.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-128. + 128. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 1.08e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.58e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (231. - 231. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 3.08e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-609. - 609. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.84e3 + 1.84e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (1.09e3 + 1.09e3i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 3.30e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 4.44e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + (-3.41e3 + 3.41e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 633.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.74e3 - 2.74e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 6.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (6.01e3 + 6.01e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 7.19e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.14e4 - 1.14e4i)T + 8.85e7iT^{2} \) |
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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
−14.61919148445916756425762172487, −13.31815449677356098880635116893, −11.90172159049490584355832729242, −11.66546195780068595860568785099, −10.31981653823407821509989249279, −8.376373553867956358527773995941, −6.59867098414694377368069909006, −5.08494792648709342496601005211, −2.96633617712367631365423643572, −0.51755799052749847032000051933,
4.37617487846356255955254197458, 4.88830655031114109273500050676, 6.93338543776440440466759244696, 8.206172414886197574287954474059, 9.822232750510355549197119980518, 11.49105644213730776227520577975, 12.41043090791025987624509221644, 13.91851052736934722388705811506, 15.21159397292360930650375246734, 15.93900416104495437125072557684
