| L(s) = 1 | − 59.7i·5-s − 483. i·7-s + 1.41e3·11-s − 3.45e3i·13-s + 3.05e3·17-s − 968.·19-s + 3.31e3i·23-s + 1.20e4·25-s − 2.63e4i·29-s + 2.71e4i·31-s − 2.88e4·35-s − 3.60e4i·37-s + 6.86e3·41-s − 9.28e4·43-s + 1.59e5i·47-s + ⋯ |
| L(s) = 1 | − 0.477i·5-s − 1.40i·7-s + 1.06·11-s − 1.57i·13-s + 0.622·17-s − 0.141·19-s + 0.272i·23-s + 0.771·25-s − 1.08i·29-s + 0.909i·31-s − 0.673·35-s − 0.712i·37-s + 0.0995·41-s − 1.16·43-s + 1.53i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.973679193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.973679193\) |
| \(L(4)\) |
|
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 | \( 1 + 59.7iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 483. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.41e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.45e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.05e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 968.T + 4.70e7T^{2} \) |
| 23 | \( 1 - 3.31e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.63e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.71e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 3.60e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.86e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 9.28e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 8.66e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.28e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 1.89e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.19e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.96e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 6.39e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.64e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 8.02e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 5.41e4T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.10e6T + 8.32e11T^{2} \) |
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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
−10.38312003951809028252649230867, −9.615979017471530217662236316148, −8.390891739112567845837892411258, −7.55966152506237938023848016718, −6.54830520922983211387925955543, −5.30619687157001469148431330188, −4.17348078199762306577968736533, −3.21385566316221497743027290598, −1.30632176824411736691524384835, −0.52116450613897478240372024023,
1.49094076126919347108344329792, 2.59568833047122304262931973103, 3.87098845637548448635301540367, 5.16683806506244870592248408079, 6.33065599553243754938313119554, 6.99655210168604422651097698009, 8.546376679586051998972263355539, 9.104635538344622237091916607743, 10.07223647921986602564525529825, 11.44853042263735759644951364008
