| L(s) = 1 | + (−4.52 + 2.55i)3-s + (−1.73 − 1.45i)5-s + (−5.32 − 1.93i)7-s + (13.9 − 23.0i)9-s + (19.5 − 16.4i)11-s + (6.38 − 36.2i)13-s + (11.5 + 2.16i)15-s + (49.6 − 85.9i)17-s + (−15.2 − 26.4i)19-s + (29.0 − 4.80i)21-s + (55.1 − 20.0i)23-s + (−20.8 − 118. i)25-s + (−4.41 + 140. i)27-s + (−3.73 − 21.2i)29-s + (−226. + 82.3i)31-s + ⋯ |
| L(s) = 1 | + (−0.871 + 0.490i)3-s + (−0.155 − 0.130i)5-s + (−0.287 − 0.104i)7-s + (0.518 − 0.855i)9-s + (0.536 − 0.450i)11-s + (0.136 − 0.772i)13-s + (0.199 + 0.0373i)15-s + (0.707 − 1.22i)17-s + (−0.184 − 0.319i)19-s + (0.301 − 0.0499i)21-s + (0.499 − 0.181i)23-s + (−0.166 − 0.944i)25-s + (−0.0315 + 0.999i)27-s + (−0.0239 − 0.135i)29-s + (−1.31 + 0.477i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.763977 - 0.519899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.763977 - 0.519899i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.52 - 2.55i)T \) |
| good | 5 | \( 1 + (1.73 + 1.45i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (5.32 + 1.93i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (-19.5 + 16.4i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-6.38 + 36.2i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-49.6 + 85.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (15.2 + 26.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-55.1 + 20.0i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (3.73 + 21.2i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (226. - 82.3i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (23.8 - 41.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-16.7 + 95.0i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (124. - 104. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-323. - 117. i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 343.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (212. + 178. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (327. + 119. i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (78.4 - 445. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-407. + 705. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-71.6 - 124. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (155. + 884. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-81.0 - 459. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (218. + 378. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.10e3 + 925. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.80578294917975440885233252182, −11.89883360587930062553545819790, −10.95078169732527817577401661370, −9.948474232328433072723057177426, −8.871378709190726123440317802557, −7.27815306413868344673910055621, −6.03938512270581466221513351277, −4.88452632568686993769752057508, −3.40699141039874618900490334626, −0.60303878722319223499554202096,
1.62271136094145334688816811467, 3.94053169129942712684116360437, 5.54298657925078843455594857448, 6.63532306972844062242223095688, 7.65645956936155258609247295804, 9.181493897712820309406944771418, 10.44065120114028096027623428909, 11.43181757324761339036313229526, 12.35118206079128604878667420861, 13.14966617475234027979069136322
