| L(s) = 1 | + 8i·2-s − 44.3i·3-s − 64·4-s + 354.·6-s + 343i·7-s − 512i·8-s + 224.·9-s + 8.29e3·11-s + 2.83e3i·12-s − 2.08e3i·13-s − 2.74e3·14-s + 4.09e3·16-s + 2.22e4i·17-s + 1.79e3i·18-s − 2.26e4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.947i·3-s − 0.5·4-s + 0.669·6-s + 0.377i·7-s − 0.353i·8-s + 0.102·9-s + 1.87·11-s + 0.473i·12-s − 0.263i·13-s − 0.267·14-s + 0.250·16-s + 1.09i·17-s + 0.0725i·18-s − 0.759·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.259453856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259453856\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 343iT \) |
| good | 3 | \( 1 + 44.3iT - 2.18e3T^{2} \) |
| 11 | \( 1 - 8.29e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.08e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.22e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.26e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.18e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 6.93e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.46e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.01e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 1.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.60e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 2.82e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 4.13e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 9.61e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.33e5iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 5.71e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.86e4iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 7.61e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.64e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 7.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.84e6iT - 8.07e13T^{2} \) |
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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
−10.31118791923473575290612264993, −9.243590841626325141018499463200, −8.431670629638588966278902143460, −7.54289357607288915225190639273, −6.44191220653227920277744851604, −6.19720654810147334139903902891, −4.62075966730449006153952891588, −3.55784632542891110961796891959, −1.82877078736798975723853622916, −1.01831747274337336092111507887,
0.56758273295647911442693825161, 1.72552978512923968753927950926, 3.19387196918847141705313523181, 4.21374631131315588638288775518, 4.65005456478097585417163664329, 6.23511386342487735867427030284, 7.24788958870918122534107285312, 8.808772649103239533336339247016, 9.295221516155015982022666940348, 10.17570008379249818267743663455
