| L(s) = 1 | + (1.30 + 1.51i)2-s + (2.52 + 1.62i)3-s + (−0.569 + 3.95i)4-s + (−4.97 + 10.9i)5-s + (0.853 + 5.93i)6-s + (10.5 + 3.10i)7-s + (−6.73 + 4.32i)8-s + (3.73 + 8.18i)9-s + (−23.0 + 6.75i)10-s + (12.9 − 14.9i)11-s + (−7.85 + 9.06i)12-s + (−72.7 + 21.3i)13-s + (9.14 + 20.0i)14-s + (−30.2 + 19.4i)15-s + (−15.3 − 4.50i)16-s + (9.18 + 63.8i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.445 + 0.975i)5-s + (0.0580 + 0.404i)6-s + (0.570 + 0.167i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.727 + 0.213i)10-s + (0.355 − 0.410i)11-s + (−0.189 + 0.218i)12-s + (−1.55 + 0.455i)13-s + (0.174 + 0.382i)14-s + (−0.520 + 0.334i)15-s + (−0.239 − 0.0704i)16-s + (0.131 + 0.911i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.908739 + 1.94637i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.908739 + 1.94637i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 - 1.51i)T \) |
| 3 | \( 1 + (-2.52 - 1.62i)T \) |
| 23 | \( 1 + (-81.2 - 74.5i)T \) |
| good | 5 | \( 1 + (4.97 - 10.9i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-10.5 - 3.10i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-12.9 + 14.9i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (72.7 - 21.3i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-9.18 - 63.8i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-7.20 + 50.1i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (0.816 + 5.67i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-175. + 112. i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (-80.2 - 175. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-61.5 + 134. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (302. + 194. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 - 604.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-307. - 90.2i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-604. + 177. i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-418. + 268. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (147. + 170. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-131. - 152. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (55.0 - 383. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (280. - 82.3i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (236. + 518. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-853. - 548. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-39.0 + 85.4i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−13.37150983994637338547370249592, −12.00669940459318711102478613655, −11.20372700338852944538568535294, −9.960739376944538747825934714008, −8.698884199714268861498333085875, −7.59791796980525195650855116065, −6.74999058963558284107867653820, −5.16831212666445811919887912322, −3.90346630644969784881847285873, −2.58546473846944170886054320054,
0.927796907904721397799601605091, 2.61363645676384229926876805114, 4.34415748876267142979654255619, 5.16659678452044056952732877007, 7.07244746644249892814902235786, 8.133233576099248705000553301450, 9.249013034623510076315940955284, 10.27848589743269996684547378521, 11.78150301259965720778028917276, 12.30222893097166303918468672572
