| L(s) = 1 | + (−1.99 − 1.44i)2-s + (−0.165 + 0.509i)3-s + (−0.595 − 1.83i)4-s + (1.24 − 0.902i)5-s + (1.06 − 0.776i)6-s + (−8.72 − 26.8i)7-s + (−7.55 + 23.2i)8-s + (21.6 + 15.7i)9-s − 3.78·10-s + 1.03·12-s + (−55.3 − 40.2i)13-s + (−21.5 + 66.2i)14-s + (0.254 + 0.782i)15-s + (36.2 − 26.3i)16-s + (−44.7 + 32.5i)17-s + (−20.3 − 62.6i)18-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.512i)2-s + (−0.0318 + 0.0980i)3-s + (−0.0744 − 0.229i)4-s + (0.111 − 0.0807i)5-s + (0.0726 − 0.0528i)6-s + (−0.471 − 1.45i)7-s + (−0.334 + 1.02i)8-s + (0.800 + 0.581i)9-s − 0.119·10-s + 0.0248·12-s + (−1.18 − 0.858i)13-s + (−0.410 + 1.26i)14-s + (0.00437 + 0.0134i)15-s + (0.567 − 0.411i)16-s + (−0.638 + 0.464i)17-s + (−0.266 − 0.819i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0754778 + 0.274429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0754778 + 0.274429i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (1.99 + 1.44i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (0.165 - 0.509i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 0.902i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (8.72 + 26.8i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (55.3 + 40.2i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (44.7 - 32.5i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (17.0 - 52.4i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-34.9 - 107. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (57.2 + 41.6i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (65.0 + 200. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-59.3 + 182. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 208.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-158. + 487. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-303. - 220. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (156. + 481. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-379. + 275. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-318. + 231. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-89.4 - 275. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-137. - 99.6i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-245. + 178. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (519. + 377. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−12.34540577902273934762568203941, −10.80952965642836890398263621596, −10.26514821655949190611619391809, −9.635019096220759988240783948589, −8.089065808008981408183493123663, −7.09208176169570766288990236883, −5.41184345114005326779803418549, −3.97669517237046364370192478555, −1.88497855500668893273697336733, −0.18082174694132606204805995383,
2.48253378755663771287027742699, 4.35613683702440141415796941300, 6.20243439271171807491121659020, 7.01698844805018158447396348763, 8.328218522642536460730558215762, 9.374186068608865504407074006679, 9.855754904757113229416629799020, 11.93410476432192405919944225600, 12.28732923166590034702909453240, 13.47524042166497500411841659582
