| L(s) = 1 | + 4.73·2-s − 41.5·4-s − 38.4i·5-s − 655. i·7-s − 499.·8-s − 181. i·10-s + 589. i·11-s − 3.17e3·13-s − 3.10e3i·14-s + 292.·16-s + 909. i·17-s + 3.91e3i·19-s + 1.59e3i·20-s + 2.79e3i·22-s + (−2.67e3 − 1.18e4i)23-s + ⋯ |
| L(s) = 1 | + 0.591·2-s − 0.649·4-s − 0.307i·5-s − 1.91i·7-s − 0.976·8-s − 0.181i·10-s + 0.443i·11-s − 1.44·13-s − 1.13i·14-s + 0.0714·16-s + 0.185i·17-s + 0.570i·19-s + 0.199i·20-s + 0.262i·22-s + (−0.220 − 0.975i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3982249088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3982249088\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (2.67e3 + 1.18e4i)T \) |
| good | 2 | \( 1 - 4.73T + 64T^{2} \) |
| 5 | \( 1 + 38.4iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 655. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 589. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 3.17e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 909. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.91e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 - 3.80e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.25e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 8.08e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.52e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 4.68e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.47e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.54e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 9.88e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 1.49e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 3.31e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.71e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 3.15e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 2.42e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 8.27e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.03e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 8.33e5iT - 8.32e11T^{2} \) |
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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
−11.88541589756369776408290334601, −10.30221278618191164251793209185, −9.965563891713532924502914707131, −8.549292388851201887001096318823, −7.48281327993029006061633420307, −6.46757149298810790454591698493, −4.73790758645870823867777590114, −4.47226738871396122087079321627, −3.09019160544593656879795812376, −1.07990139613682080931734622466,
0.10077571899656979686990521308, 2.33922805971709195637005116352, 3.24436466712244852634832388438, 4.93156679555005086911378246448, 5.47282854259483870960860629333, 6.67522141182617114657696765090, 8.230148843315828502175085357293, 9.089979271791479246961314229140, 9.813938831492950837803579345200, 11.37939802262335276016420516087
