| L(s) = 1 | − 12.4·2-s + 30.2i·3-s + 90.5·4-s + 96.4·5-s − 376. i·6-s + 161.·7-s − 329.·8-s − 186.·9-s − 1.19e3·10-s + 701. i·11-s + 2.73e3i·12-s + 3.27e3i·13-s − 2.01e3·14-s + 2.91e3i·15-s − 1.69e3·16-s − 2.15e3i·17-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 1.12i·3-s + 1.41·4-s + 0.771·5-s − 1.74i·6-s + 0.472·7-s − 0.643·8-s − 0.256·9-s − 1.19·10-s + 0.526i·11-s + 1.58i·12-s + 1.49i·13-s − 0.733·14-s + 0.864i·15-s − 0.413·16-s − 0.438i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.412722 + 0.701605i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412722 + 0.701605i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-1.44e4 - 2.60e4i)T \) |
| good | 2 | \( 1 + 12.4T + 64T^{2} \) |
| 3 | \( 1 - 30.2iT - 729T^{2} \) |
| 5 | \( 1 - 96.4T + 1.56e4T^{2} \) |
| 7 | \( 1 - 161.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 701. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 3.27e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 2.15e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 3.21e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.47e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.94e3iT - 5.94e8T^{2} \) |
| 37 | \( 1 - 8.17e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.28e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 4.20e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 1.54e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.00e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 6.40e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 3.87e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 9.93e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.17e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 6.15e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 1.02e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.80e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 4.50e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 3.41e5T + 8.32e11T^{2} \) |
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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
−16.39855912511606682384309066878, −15.24723389480209312129656424954, −13.84575780778311828610270337141, −11.63274255503057568179392777400, −10.34757251119711660680782705817, −9.638607185253847529640303502333, −8.651436983599394536510246554993, −6.86544742281649385520769462726, −4.63477675449427703599916416220, −1.81614438312765392296263881268,
0.75062963100294599963156124590, 2.07599956034461211102338320080, 6.00646848803736778951684018075, 7.52432269567812081999268666306, 8.416242987238362881357558377213, 9.920238483644613788106180673262, 11.04995932791456432839842622219, 12.68914280645134550333005946325, 13.79718984969327008292359499839, 15.53352813637390124250743798310
