| L(s) = 1 | + i·2-s − 4-s + 2.46·5-s + (−1.59 + 2.11i)7-s − i·8-s + 2.46i·10-s + 2.62i·11-s − 2.71i·13-s + (−2.11 − 1.59i)14-s + 16-s − 4.00·17-s + 8.27i·19-s − 2.46·20-s − 2.62·22-s + 1.61i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.10·5-s + (−0.601 + 0.799i)7-s − 0.353i·8-s + 0.778i·10-s + 0.792i·11-s − 0.753i·13-s + (−0.565 − 0.425i)14-s + 0.250·16-s − 0.972·17-s + 1.89i·19-s − 0.550·20-s − 0.560·22-s + 0.336i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.381452455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381452455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.59 - 2.11i)T \) |
| good | 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 - 2.62iT - 11T^{2} \) |
| 13 | \( 1 + 2.71iT - 13T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 - 8.27iT - 19T^{2} \) |
| 23 | \( 1 - 1.61iT - 23T^{2} \) |
| 29 | \( 1 - 1.98iT - 29T^{2} \) |
| 31 | \( 1 - 0.851iT - 31T^{2} \) |
| 37 | \( 1 - 11.9T + 37T^{2} \) |
| 41 | \( 1 + 5.52T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.19iT - 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 - 8.74iT - 61T^{2} \) |
| 67 | \( 1 + 0.461T + 67T^{2} \) |
| 71 | \( 1 - 7.30iT - 71T^{2} \) |
| 73 | \( 1 - 0.0359iT - 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 1.80iT - 97T^{2} \) |
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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
−9.800100635923405119243323696607, −9.472159780193203699820599166556, −8.486047896363703968028218124200, −7.67250312048388335451050242589, −6.58740093052531254668606358423, −5.93818397447764617220598746353, −5.38345918885620747229513731144, −4.20273275178591688171904681633, −2.87236786487325537932402539625, −1.73178340295863690539930941498,
0.58791818928482422227081426432, 2.06904124518590419487035568806, 2.98286096702101385901142785798, 4.17617993395316103507701257837, 5.00776054137196765381827924298, 6.29664531628840142661186708295, 6.68657387329370158397957580327, 7.988526397456486560632659594402, 9.181956023622811809835811136553, 9.388576584966175121879243744965
