| L(s) = 1 | + (−65.3 − 47.8i)3-s + (−529. + 332. i)5-s − 1.22e3i·7-s + (1.98e3 + 6.25e3i)9-s + 2.74e4i·11-s + 3.85e4i·13-s + (5.05e4 + 3.55e3i)15-s − 1.42e5·17-s + 1.65e5·19-s + (−5.87e4 + 8.02e4i)21-s + 1.46e5·23-s + (1.69e5 − 3.52e5i)25-s + (1.69e5 − 5.03e5i)27-s + 3.16e4i·29-s + 1.95e5·31-s + ⋯ |
| L(s) = 1 | + (−0.806 − 0.590i)3-s + (−0.846 + 0.532i)5-s − 0.511i·7-s + (0.302 + 0.953i)9-s + 1.87i·11-s + 1.34i·13-s + (0.997 + 0.0702i)15-s − 1.71·17-s + 1.27·19-s + (−0.302 + 0.412i)21-s + 0.521·23-s + (0.432 − 0.901i)25-s + (0.318 − 0.947i)27-s + 0.0447i·29-s + 0.211·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0702i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.997 - 0.0702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6075886923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6075886923\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (65.3 + 47.8i)T \) |
| 5 | \( 1 + (529. - 332. i)T \) |
| good | 7 | \( 1 + 1.22e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.74e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.85e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.42e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.65e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.46e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 3.16e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.95e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.15e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 4.37e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.97e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.90e5T + 2.38e13T^{2} \) |
| 53 | \( 1 - 7.12e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 4.09e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.39e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.93e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.51e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.32e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.71e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 1.71e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 2.41e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 6.69e7iT - 7.83e15T^{2} \) |
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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
−11.42436230219185201232384796060, −10.39915858338999344487335478396, −9.356757416467172044053700493283, −7.84541994747696691216543367183, −6.95937744562194358199235286579, −6.67387806977281364027355727888, −4.79573577181254721713767823021, −4.23907875138617175756616725421, −2.43659104466525116875355428085, −1.28032467472882136040891324205,
0.22024081005214716085574094440, 0.802593561822052273288882832338, 3.00965837913132638281003516384, 3.95680033582855454662798613027, 5.25527296989499548499290496688, 5.80487104976670470141720110567, 7.20600992446504179716397875294, 8.545137302330256182012927979689, 9.035118953213457910575790196099, 10.52514658961941707685663669943
