| L(s) = 1 | + (51.8 − 62.2i)3-s + (254. − 570. i)5-s + 4.20e3i·7-s + (−1.17e3 − 6.45e3i)9-s + 542. i·11-s + 4.07e4i·13-s + (−2.22e4 − 4.54e4i)15-s − 3.12e4·17-s − 2.30e5·19-s + (2.61e5 + 2.18e5i)21-s − 4.65e5·23-s + (−2.60e5 − 2.90e5i)25-s + (−4.62e5 − 2.61e5i)27-s + 2.31e5i·29-s − 2.06e5·31-s + ⋯ |
| L(s) = 1 | + (0.640 − 0.767i)3-s + (0.407 − 0.913i)5-s + 1.75i·7-s + (−0.179 − 0.983i)9-s + 0.0370i·11-s + 1.42i·13-s + (−0.440 − 0.897i)15-s − 0.373·17-s − 1.77·19-s + (1.34 + 1.12i)21-s − 1.66·23-s + (−0.667 − 0.744i)25-s + (−0.870 − 0.492i)27-s + 0.326i·29-s − 0.223·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7663782210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7663782210\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-51.8 + 62.2i)T \) |
| 5 | \( 1 + (-254. + 570. i)T \) |
| good | 7 | \( 1 - 4.20e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 542. iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.07e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 3.12e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.30e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.65e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 2.31e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 2.06e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 2.73e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.74e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.38e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 2.05e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.09e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 5.20e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.93e3T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.61e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 4.61e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.27e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 9.49e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 6.39e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 4.56e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 2.57e7iT - 7.83e15T^{2} \) |
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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
−12.28682162127663487994696214693, −11.71606259605016425433335443978, −9.736299706718607444359224027795, −8.716033806671916859734297546525, −8.462817018130882550734257517792, −6.63106346350932101722543348567, −5.76166161107410075884380105855, −4.24473040024158807569577138666, −2.30000990339801981073708852462, −1.78832057659707658561008972504,
0.16400899401940619329847603607, 2.14620721237045854855661504439, 3.48856676737200357400694779880, 4.34260440448693219644802029172, 6.03649795521673664981294980475, 7.36878471765350688729609692376, 8.241325311699589977995845061645, 9.833991828216109468004639206871, 10.47927366678732769068069749590, 10.97137656380336311863334606989
