| L(s) = 1 | + (12.4 − 10i)2-s − 99.9·3-s + (56 − 249. i)4-s + (−1.24e3 + 999. i)6-s + 1.39e3·7-s + (−1.79e3 − 3.68e3i)8-s + 3.42e3·9-s + 1.84e4i·11-s + (−5.59e3 + 2.49e4i)12-s + 5.47e3i·13-s + (1.74e4 − 1.39e4i)14-s + (−5.92e4 − 2.79e4i)16-s + 7.30e4i·17-s + (4.27e4 − 3.42e4i)18-s − 1.94e4i·19-s + ⋯ |
| L(s) = 1 | + (0.780 − 0.625i)2-s − 1.23·3-s + (0.218 − 0.975i)4-s + (−0.962 + 0.770i)6-s + 0.582·7-s + (−0.439 − 0.898i)8-s + 0.521·9-s + 1.26i·11-s + (−0.269 + 1.20i)12-s + 0.191i·13-s + (0.454 − 0.364i)14-s + (−0.904 − 0.426i)16-s + 0.875i·17-s + (0.407 − 0.326i)18-s − 0.149i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.90541 - 0.678500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90541 - 0.678500i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-12.4 + 10i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 99.9T + 6.56e3T^{2} \) |
| 7 | \( 1 - 1.39e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.84e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 5.47e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 7.30e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.94e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.37e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.28e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 6.79e4iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.47e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 2.14e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 5.92e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 7.62e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 8.24e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 3.72e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.47e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.52e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.19e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.72e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 3.59e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 5.19e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 8.33e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.20e8iT - 7.83e15T^{2} \) |
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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
−12.25225481705429538781857508678, −11.11043805322940385782350800311, −10.59782917104236340226055044393, −9.261400447648240787522565066430, −7.28130181556114174678556267424, −6.08560739930847843434080203054, −5.08320770220900098195877677616, −4.15543173806726832947544351467, −2.20001888641156286560004395033, −0.872563177910579807577212355639,
0.73561387542887190921855179860, 2.97334499042874556738589683028, 4.61221185586976884344238850829, 5.52685017118081355350090211770, 6.37017927208305022465075357903, 7.64163071005776073308637645796, 8.860597398516756354037583530501, 10.80386081314715990633288611430, 11.45832858951779295515523399615, 12.29215958328603789289334640537
