| L(s) = 1 | + 214. i·5-s + 616.·7-s + 5.12e3i·11-s − 39.5i·13-s − 3.77e3·17-s − 2.12e4i·19-s + 2.41e4·23-s + 3.22e4·25-s + 1.64e5i·29-s + 2.14e5·31-s + 1.32e5i·35-s − 7.67e4i·37-s + 2.08e5·41-s + 5.96e5i·43-s + 3.03e5·47-s + ⋯ |
| L(s) = 1 | + 0.766i·5-s + 0.678·7-s + 1.16i·11-s − 0.00499i·13-s − 0.186·17-s − 0.709i·19-s + 0.413·23-s + 0.412·25-s + 1.25i·29-s + 1.29·31-s + 0.520i·35-s − 0.249i·37-s + 0.471·41-s + 1.14i·43-s + 0.427·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.360703794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360703794\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 214. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 616.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.12e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 + 39.5iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 3.77e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.12e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.41e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.64e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.67e4iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 2.08e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.96e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 3.03e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.73e4iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 6.22e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.89e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 3.20e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 4.09e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.52e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 1.08e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.10e7T + 8.07e13T^{2} \) |
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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
−9.910537753007487126679319324734, −9.017653295851025891194346335110, −7.989891518082642884851406550525, −7.11403316743546650710560777272, −6.49291035830447366843627636156, −5.10837709430538833911370839989, −4.43774350614545811787220734872, −3.09679608964965850092848549217, −2.19097246942979421054913178777, −1.05810400553615557685768833790,
0.49578418999686421563825969509, 1.27891714276018275772213092128, 2.53917011496576101443087458143, 3.79113884292515341053720765770, 4.76346555213741410128226635683, 5.60969443803752932696118706462, 6.53488151714982351103195491705, 7.86786045658766416303554168276, 8.402104867786523011649666409234, 9.180512908911820280444085135011
