| L(s) = 1 | + (−31.2 + 18.0i)3-s + (−305. − 176. i)5-s + (−2.23e3 − 872. i)7-s + (−2.63e3 + 4.55e3i)9-s + (−6.59e3 − 1.14e4i)11-s − 2.53e4i·13-s + 1.27e4·15-s + (−1.29e5 + 7.48e4i)17-s + (8.86e4 + 5.12e4i)19-s + (8.55e4 − 1.30e4i)21-s + (7.99e4 − 1.38e5i)23-s + (−1.33e5 − 2.30e5i)25-s − 4.26e5i·27-s + 6.51e5·29-s + (−6.99e5 + 4.03e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.385 + 0.222i)3-s + (−0.488 − 0.281i)5-s + (−0.931 − 0.363i)7-s + (−0.400 + 0.694i)9-s + (−0.450 − 0.780i)11-s − 0.887i·13-s + 0.250·15-s + (−1.55 + 0.895i)17-s + (0.680 + 0.392i)19-s + (0.439 − 0.0672i)21-s + (0.285 − 0.494i)23-s + (−0.341 − 0.590i)25-s − 0.801i·27-s + 0.920·29-s + (−0.757 + 0.437i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7289086485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7289086485\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.23e3 + 872. i)T \) |
| good | 3 | \( 1 + (31.2 - 18.0i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (305. + 176. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (6.59e3 + 1.14e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 2.53e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.29e5 - 7.48e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-8.86e4 - 5.12e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-7.99e4 + 1.38e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 6.51e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (6.99e5 - 4.03e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (5.24e5 - 9.08e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.60e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.42e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.13e6 - 4.12e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.30e6 - 7.45e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-2.41e5 + 1.39e5i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-3.61e6 - 2.08e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.64e7 - 2.85e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.27e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-4.46e7 + 2.58e7i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.54e7 + 2.67e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.80e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.47e7 - 8.51e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.20e8iT - 7.83e15T^{2} \) |
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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
−12.16147197112410618640547480182, −10.85750968688859781187793457271, −10.36306774065324211236554841472, −8.794858591859365325747633436827, −7.893990838070186276086007438170, −6.44826164822460207117651949967, −5.34864584193486725239057211423, −4.00951012323962801734610904309, −2.69228905200288231354857439042, −0.59244514211452430762697864210,
0.38003877637891352629227066053, 2.35040225366630296516024610269, 3.66823872934660068845318187625, 5.16799013893192042891117861540, 6.60107141760846891489908837323, 7.19902240349730807299336413249, 8.941918564361182731406795254919, 9.666061656951519521118606118759, 11.20790164250461171825617792844, 11.84986905484463635169149357246
