| L(s) = 1 | + (−11.0 + 11.5i)2-s − 27.2·3-s + (−11.6 − 255. i)4-s + (301. − 315. i)6-s − 3.32e3·7-s + (3.08e3 + 2.69e3i)8-s − 5.81e3·9-s + 6.36e3i·11-s + (316. + 6.96e3i)12-s + 3.07e4i·13-s + (3.67e4 − 3.84e4i)14-s + (−6.52e4 + 5.94e3i)16-s + 1.22e5i·17-s + (6.43e4 − 6.73e4i)18-s + 7.55e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.690 + 0.722i)2-s − 0.336·3-s + (−0.0453 − 0.998i)4-s + (0.232 − 0.243i)6-s − 1.38·7-s + (0.753 + 0.657i)8-s − 0.886·9-s + 0.434i·11-s + (0.0152 + 0.335i)12-s + 1.07i·13-s + (0.956 − 1.00i)14-s + (−0.995 + 0.0906i)16-s + 1.46i·17-s + (0.612 − 0.641i)18-s + 0.0579i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.249962 - 0.0530481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.249962 - 0.0530481i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.0 - 11.5i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 27.2T + 6.56e3T^{2} \) |
| 7 | \( 1 + 3.32e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.36e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.07e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.22e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.55e3iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.06e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 8.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.39e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.86e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 9.72e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.61e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 1.87e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 8.01e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 6.18e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.79e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.19e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.62e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.35e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 1.22e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 6.96e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 5.58e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.97e7iT - 7.83e15T^{2} \) |
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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
−12.12290417970799851128570090499, −10.89365952956292312161437121249, −9.807728288068449253659081051034, −9.009483697344171640520736271252, −7.72232800250094965083674901172, −6.37616168176878310893745802385, −5.86543513388531925084299873733, −4.02746508032004860695432626409, −2.04680612039142073391134440942, −0.16884281029342925869163279482,
0.59943343196546678345008154492, 2.65864781184241846450353329403, 3.51572409690529062436029390446, 5.50006662237567429951953389518, 6.82844622902610911090566442954, 8.164577741571636642733862646808, 9.301409755108810032905550518631, 10.16589663651559471697566484831, 11.22520689236834646260845898189, 12.17811380057807532828252417792
