| L(s) = 1 | + 7.19i·2-s + 44.5·3-s + 12.2·4-s + 52.8·5-s + 320. i·6-s − 372.·7-s + 548. i·8-s + 1.25e3·9-s + 380. i·10-s + 2.14e3i·11-s + 545.·12-s − 2.57e3i·13-s − 2.68e3i·14-s + 2.35e3·15-s − 3.16e3·16-s + 6.45e3·17-s + ⋯ |
| L(s) = 1 | + 0.899i·2-s + 1.64·3-s + 0.191·4-s + 0.422·5-s + 1.48i·6-s − 1.08·7-s + 1.07i·8-s + 1.71·9-s + 0.380i·10-s + 1.61i·11-s + 0.315·12-s − 1.17i·13-s − 0.977i·14-s + 0.697·15-s − 0.772·16-s + 1.31·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0784 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.52702 + 2.33586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52702 + 2.33586i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (1.61e4 - 2.04e5i)T \) |
| good | 2 | \( 1 - 7.19iT - 64T^{2} \) |
| 3 | \( 1 - 44.5T + 729T^{2} \) |
| 5 | \( 1 - 52.8T + 1.56e4T^{2} \) |
| 7 | \( 1 + 372.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 2.14e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.57e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.45e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 8.21e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 282. iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.04e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.60e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.97e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 3.55e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 1.25e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.25e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.63e5T + 2.21e10T^{2} \) |
| 61 | \( 1 - 2.07e3iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 8.63e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.13e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.22e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.95e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 8.92e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 9.53e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.62e5iT - 8.32e11T^{2} \) |
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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
−14.41699046349631564512899280490, −13.38237874173508550298074332590, −12.34399218315582363800369064316, −10.04321484634962290896088267647, −9.410615066226310294704090174598, −7.82761554848457513731917284178, −7.26197164482374990247643688104, −5.60941334388970655525126375135, −3.40183028099912807174147929079, −2.13804955975545090562474837398,
1.40432779906324150530149620729, 2.98249693097878440324157669151, 3.50892835632820820816036265078, 6.31008964894931993176933956818, 7.80133020226202297336562560166, 9.353494424190167826423012768244, 9.749452823884780017204180963314, 11.31647685402904419186164097014, 12.66071758383069082648117240467, 13.67674175519188480553877304193
