| L(s) = 1 | + (1.37 + 5.12i)2-s + (4.15 − 3.12i)3-s + (−17.4 + 10.0i)4-s + (−5.80 + 5.80i)5-s + (21.6 + 17.0i)6-s + (20.4 + 5.47i)7-s + (−45.5 − 45.5i)8-s + (7.52 − 25.9i)9-s + (−37.7 − 21.7i)10-s + (24.5 − 6.59i)11-s + (−41.0 + 96.3i)12-s + (7.53 − 46.2i)13-s + 112. i·14-s + (−6.00 + 42.2i)15-s + (90.4 − 156. i)16-s + (−12.2 − 21.1i)17-s + ⋯ |
| L(s) = 1 | + (0.485 + 1.81i)2-s + (0.799 − 0.600i)3-s + (−2.18 + 1.25i)4-s + (−0.519 + 0.519i)5-s + (1.47 + 1.15i)6-s + (1.10 + 0.295i)7-s + (−2.01 − 2.01i)8-s + (0.278 − 0.960i)9-s + (−1.19 − 0.689i)10-s + (0.674 − 0.180i)11-s + (−0.987 + 2.31i)12-s + (0.160 − 0.986i)13-s + 2.14i·14-s + (−0.103 + 0.727i)15-s + (1.41 − 2.44i)16-s + (−0.174 − 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.898513 + 1.52953i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898513 + 1.52953i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.15 + 3.12i)T \) |
| 13 | \( 1 + (-7.53 + 46.2i)T \) |
| good | 2 | \( 1 + (-1.37 - 5.12i)T + (-6.92 + 4i)T^{2} \) |
| 5 | \( 1 + (5.80 - 5.80i)T - 125iT^{2} \) |
| 7 | \( 1 + (-20.4 - 5.47i)T + (297. + 171.5i)T^{2} \) |
| 11 | \( 1 + (-24.5 + 6.59i)T + (1.15e3 - 665.5i)T^{2} \) |
| 17 | \( 1 + (12.2 + 21.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-3.46 + 12.9i)T + (-5.94e3 - 3.42e3i)T^{2} \) |
| 23 | \( 1 + (80.7 - 139. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (170. + 98.4i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (72.8 + 72.8i)T + 2.97e4iT^{2} \) |
| 37 | \( 1 + (-39.5 - 147. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (75.8 + 282. i)T + (-5.96e4 + 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-227. + 131. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-24.7 - 24.7i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 - 380. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-10.5 + 39.2i)T + (-1.77e5 - 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-272. - 471. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (102. - 27.4i)T + (2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-161. - 43.3i)T + (3.09e5 + 1.78e5i)T^{2} \) |
| 73 | \( 1 + (321. - 321. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 156.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-252. + 252. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + (-658. + 176. i)T + (6.10e5 - 3.52e5i)T^{2} \) |
| 97 | \( 1 + (228. - 852. i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.55572982006111298294648152245, −15.02693596210569552473861622116, −14.16027108736063643509856310238, −13.21985804618784166988733616872, −11.73288274855221940444122397342, −9.104971677465188559942045394066, −7.952014218515582840214590695437, −7.29300303211385538082671428117, −5.71327190969502342710927777367, −3.79707725623845245746620708055,
1.82882298160697658490823140556, 3.94163760414396789402862209017, 4.68646160301072874389055754271, 8.339785981958503264689214432521, 9.351286521213793018453135866875, 10.69931709930923482019566029224, 11.61752086084867001527076440809, 12.78955191843304816925740178682, 14.18729337574123701808922233878, 14.60474865258939185314624615681
