| L(s) = 1 | + 5.65i·2-s − 32.0·4-s + 173. i·5-s − 484·7-s − 181. i·8-s − 984.·10-s + 1.34e3i·11-s + 3.36e3·13-s − 2.73e3i·14-s + 1.02e3·16-s + 12.7i·17-s + 5.74e3·19-s − 5.56e3i·20-s − 7.58e3·22-s − 3.37e3i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 1.39i·5-s − 1.41·7-s − 0.353i·8-s − 0.984·10-s + 1.00i·11-s + 1.53·13-s − 0.997i·14-s + 0.250·16-s + 0.00259i·17-s + 0.837·19-s − 0.695i·20-s − 0.712·22-s − 0.277i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.329918 + 1.03801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.329918 + 1.03801i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 173. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 484T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.34e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.36e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.37e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 2.93e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 3.97e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.70e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 3.80e3T + 6.32e9T^{2} \) |
| 47 | \( 1 - 7.67e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.49e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.32e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.68e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.51e4T + 2.43e11T^{2} \) |
| 83 | \( 1 + 1.09e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.21e5T + 8.32e11T^{2} \) |
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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
−18.06910386729150301377131706949, −16.32436931584549506497081861955, −15.37463139302735411915003575388, −14.12812289850175296566168762207, −12.79943948634922932546375385271, −10.75716920754941246673460170820, −9.404076889061244728090908165569, −7.23424034812212747387496025669, −6.19948274389871141938995126915, −3.40488705269129499046290458171,
0.77636111964462645662059400391, 3.63132619809715011724446302242, 5.80918410576232217533082322387, 8.549264004115012358299792564715, 9.630622809509253975325652294077, 11.43224405226010532435699155407, 12.91917231506903117264513196133, 13.51344595741453586559357503664, 15.96786541734827632151247851394, 16.62029485204699025951331168978
