| L(s) = 1 | + (−23.4 + 13.3i)3-s − 100. i·5-s − 56.7·7-s + (369. − 628. i)9-s + 1.56e3i·11-s + 2.79e3·13-s + (1.34e3 + 2.34e3i)15-s + 7.75e3i·17-s + 1.05e4·19-s + (1.33e3 − 760. i)21-s − 1.84e4i·23-s + 5.59e3·25-s + (−255. + 1.96e4i)27-s + 2.08e4i·29-s + 3.28e3·31-s + ⋯ |
| L(s) = 1 | + (−0.868 + 0.496i)3-s − 0.801i·5-s − 0.165·7-s + (0.507 − 0.861i)9-s + 1.17i·11-s + 1.27·13-s + (0.397 + 0.695i)15-s + 1.57i·17-s + 1.53·19-s + (0.143 − 0.0821i)21-s − 1.51i·23-s + 0.358·25-s + (−0.0129 + 0.999i)27-s + 0.856i·29-s + 0.110·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.28203 + 0.340551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28203 + 0.340551i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (23.4 - 13.3i)T \) |
| good | 5 | \( 1 + 100. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 56.7T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.56e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.79e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.75e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.05e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.84e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.08e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.28e3T + 8.87e8T^{2} \) |
| 37 | \( 1 - 3.93e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 4.46e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 2.36e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 6.50e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 6.57e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 3.62e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 4.12e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.65e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 5.52e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.24e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.81e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 2.49e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 3.07e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.49e6T + 8.32e11T^{2} \) |
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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
−14.72437882684485328752472373541, −12.97233496419128778675580973210, −12.26813173586572790287032657837, −10.91311366242699595566403167047, −9.818194187092864340830954430329, −8.527292938315063779284928233803, −6.65303084230754778779926859427, −5.28159273344982012771145042059, −3.99465763948517162967677950245, −1.14340361775573487873736999239,
0.893452686496486292142571378422, 3.19557504539974704825985499660, 5.44530452595381994521676310946, 6.57589864664399616294330889710, 7.80870639853819742932064621262, 9.652375527494184500874440609856, 11.22237360681758328425976921166, 11.52512747676589396130705723012, 13.34916833562778729704915831828, 13.95154862605357309594815547330
