| L(s) = 1 | + 5.23i·2-s − 5.47i·3-s + 100.·4-s + 28.6·6-s − 769. i·7-s + 1.19e3i·8-s + 2.15e3·9-s + 5.35e3·11-s − 550. i·12-s + 1.40e4i·13-s + 4.03e3·14-s + 6.60e3·16-s − 1.24e4i·17-s + 1.12e4i·18-s + 5.03e3·19-s + ⋯ |
| L(s) = 1 | + 0.462i·2-s − 0.117i·3-s + 0.785·4-s + 0.0542·6-s − 0.848i·7-s + 0.826i·8-s + 0.986·9-s + 1.21·11-s − 0.0919i·12-s + 1.77i·13-s + 0.392·14-s + 0.402·16-s − 0.612i·17-s + 0.456i·18-s + 0.168·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.11423 + 0.499103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.11423 + 0.499103i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 5.23iT - 128T^{2} \) |
| 3 | \( 1 + 5.47iT - 2.18e3T^{2} \) |
| 7 | \( 1 + 769. iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 5.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.40e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.24e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 5.03e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.51e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.95e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.35e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.87e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.39e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.06e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 6.26e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.14e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.57e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 8.07e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.72e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.74e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 2.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.90e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.63e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.01e7iT - 8.07e13T^{2} \) |
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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
−16.42968696656867876419634767607, −14.84682083325472718855646838966, −13.83778597006325564688980788763, −12.11981894583724627785286279476, −10.96595473521302556272383642380, −9.328172890635047776545287451185, −7.32282306089010152130969557825, −6.57041108855247341860143856537, −4.20827848772313087888391801884, −1.63010751362903130803099340907,
1.55180083548303927046301528055, 3.47736819273146495272757502644, 5.85766941666068925449908857942, 7.50438850395214607619438580543, 9.435005129300722807242883714286, 10.72386949988730731463457672693, 12.03359772519749169140734317149, 12.97545163076239425021706126179, 15.05538705931091709593352833280, 15.63305587058576950153223845110
