| L(s) = 1 | − 414. i·5-s + (1.95e3 − 1.39e3i)7-s + 2.15e4·11-s + 4.37e3i·13-s + 1.16e5i·17-s + 1.90e5i·19-s − 1.25e5·23-s + 2.19e5·25-s − 4.83e5·29-s + 2.45e5i·31-s + (−5.76e5 − 8.09e5i)35-s − 1.09e5·37-s + 3.89e6i·41-s − 1.39e6·43-s + 1.96e6i·47-s + ⋯ |
| L(s) = 1 | − 0.662i·5-s + (0.814 − 0.579i)7-s + 1.47·11-s + 0.153i·13-s + 1.40i·17-s + 1.46i·19-s − 0.448·23-s + 0.561·25-s − 0.684·29-s + 0.265i·31-s + (−0.384 − 0.539i)35-s − 0.0584·37-s + 1.37i·41-s − 0.408·43-s + 0.401i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.814 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.540975658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540975658\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.95e3 + 1.39e3i)T \) |
| good | 5 | \( 1 + 414. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 2.15e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 4.37e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.16e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.90e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.25e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 4.83e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 2.45e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.09e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.89e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.39e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 1.96e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 7.39e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.79e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.31e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 7.63e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.98e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 2.57e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.07e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 6.12e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 7.65e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.16e7iT - 7.83e15T^{2} \) |
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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
−10.72372839480907902473349342939, −9.743803867020129526314156487907, −8.642521617304677800788100336264, −7.983389716039173047076519722736, −6.71629406682311049002273856387, −5.67682475928735131646866185187, −4.37375551849731668682321333226, −3.73160425112948031818987979170, −1.69178594369843542944134638230, −1.14806886047034354907739673327,
0.59624914969025484322418053346, 1.95933975073687462789154943741, 3.03948147999387760457402739343, 4.37190981370853082909210421864, 5.44050491287973810572263303994, 6.68299761395938933523673123518, 7.40662905418295375274534939373, 8.790247115227994505017253349181, 9.360248709246096120378561444391, 10.69412949166422889840203765342
