| L(s) = 1 | + (3.14 + 1.14i)2-s + (1.66 − 2.49i)3-s + (5.51 + 4.62i)4-s + (−2.27 + 4.45i)5-s + (8.09 − 5.93i)6-s + (1.49 + 1.78i)7-s + (5.34 + 9.25i)8-s + (−3.44 − 8.31i)9-s + (−12.2 + 11.4i)10-s + (−3.16 + 0.557i)11-s + (20.7 − 6.04i)12-s + (−6.79 − 18.6i)13-s + (2.66 + 7.32i)14-s + (7.32 + 13.0i)15-s + (1.21 + 6.88i)16-s + (0.0949 − 0.164i)17-s + ⋯ |
| L(s) = 1 | + (1.57 + 0.572i)2-s + (0.555 − 0.831i)3-s + (1.37 + 1.15i)4-s + (−0.454 + 0.890i)5-s + (1.34 − 0.989i)6-s + (0.213 + 0.254i)7-s + (0.668 + 1.15i)8-s + (−0.383 − 0.923i)9-s + (−1.22 + 1.14i)10-s + (−0.287 + 0.0507i)11-s + (1.72 − 0.503i)12-s + (−0.522 − 1.43i)13-s + (0.190 + 0.522i)14-s + (0.488 + 0.872i)15-s + (0.0758 + 0.430i)16-s + (0.00558 − 0.00967i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.29143 + 0.732966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.29143 + 0.732966i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.66 + 2.49i)T \) |
| 5 | \( 1 + (2.27 - 4.45i)T \) |
| good | 2 | \( 1 + (-3.14 - 1.14i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-1.49 - 1.78i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (3.16 - 0.557i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (6.79 + 18.6i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-0.0949 + 0.164i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-12.0 - 20.9i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (17.2 + 14.4i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (17.0 - 46.7i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-14.6 - 12.2i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-24.3 - 14.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-13.0 - 35.7i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (15.6 - 2.76i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (18.7 - 15.7i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 80.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-44.9 - 7.92i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-42.9 + 36.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (43.8 + 120. i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-49.0 - 28.3i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (99.7 - 57.6i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (13.9 + 5.07i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-66.4 - 24.1i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (108. - 62.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-157. + 27.8i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.13632310391802844502030152057, −12.40432847023625184205459499802, −11.63048899287473762993988141847, −10.17274340525056393053897295539, −8.142459364808405467569638282835, −7.47436932846879915154318742555, −6.43002168303463310429648458709, −5.35004996376131777382974559196, −3.60298042181126568501173346188, −2.68164060843799989389522791679,
2.30612752993997135053536102437, 3.95301270883885748917454081639, 4.53273001028278826543736474599, 5.59874355747405169692822015896, 7.48731674233729291658438564569, 8.917910103031243981456815141100, 9.974495834328327743620313869772, 11.44865029460852526194270518483, 11.75833581668578563588340422663, 13.23301302612536741504791673834
