| L(s) = 1 | + 59.7i·3-s + 438. i·7-s − 1.38e3·9-s − 5.75e3·11-s − 3.53e3i·13-s + 2.39e4i·17-s + 1.65e4·19-s − 2.61e4·21-s + 6.56e4i·23-s + 4.80e4i·27-s − 1.34e5·29-s − 1.29e5·31-s − 3.44e5i·33-s − 1.61e5i·37-s + 2.10e5·39-s + ⋯ |
| L(s) = 1 | + 1.27i·3-s + 0.482i·7-s − 0.631·9-s − 1.30·11-s − 0.445i·13-s + 1.18i·17-s + 0.554·19-s − 0.616·21-s + 1.12i·23-s + 0.470i·27-s − 1.02·29-s − 0.777·31-s − 1.66i·33-s − 0.522i·37-s + 0.569·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1388723527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1388723527\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 59.7iT - 2.18e3T^{2} \) |
| 7 | \( 1 - 438. iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 5.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 3.53e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.39e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 1.65e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.56e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 1.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.61e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.88e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 3.43e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 1.66e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.54e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.52e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.56e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 2.99e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.12e5iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 1.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.21e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 5.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 7.20e6iT - 8.07e13T^{2} \) |
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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
−10.02021327266564090686881495300, −9.186612427021295885113330025218, −8.247394004162863239209260838354, −7.28337677361872083820982113200, −5.60277421073355091231339236556, −5.30820399879821900283407558092, −3.97835859883630166132138115744, −3.17460544010588204950825370308, −1.85159152900088429812577339675, −0.03206930814148892923898344113,
0.898225616622116224618004423504, 2.05323557351448525339766254287, 3.00523475630506274519656812089, 4.54908149208474152723578358097, 5.61723067415261720381248308101, 6.78560662261644553489780872780, 7.41538764714849426432594726885, 8.097972073282876305048015208154, 9.286521564246585792148560290487, 10.33240248262101447475358009118
