| L(s) = 1 | + (0.415 + 0.909i)2-s + (0.959 − 0.281i)3-s + (−0.654 + 0.755i)4-s + (0.273 + 0.175i)5-s + (0.654 + 0.755i)6-s + (0.346 + 2.40i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.0462 + 0.321i)10-s + (1.41 − 3.09i)11-s + (−0.415 + 0.909i)12-s + (−0.0383 + 0.266i)13-s + (−2.04 + 1.31i)14-s + (0.311 + 0.0914i)15-s + (−0.142 − 0.989i)16-s + (−3.17 − 3.66i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (0.553 − 0.162i)3-s + (−0.327 + 0.377i)4-s + (0.122 + 0.0784i)5-s + (0.267 + 0.308i)6-s + (0.130 + 0.909i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0146 + 0.101i)10-s + (0.425 − 0.932i)11-s + (−0.119 + 0.262i)12-s + (−0.0106 + 0.0739i)13-s + (−0.546 + 0.351i)14-s + (0.0804 + 0.0236i)15-s + (−0.0355 − 0.247i)16-s + (−0.770 − 0.888i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28874 + 0.621906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28874 + 0.621906i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.20 + 2.31i)T \) |
| good | 5 | \( 1 + (-0.273 - 0.175i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.346 - 2.40i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.41 + 3.09i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0383 - 0.266i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (3.17 + 3.66i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (2.63 - 3.04i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (1.31 + 1.51i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.12 - 2.67i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.69 - 1.09i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (2.19 + 1.40i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.77 - 1.10i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 + (-0.153 - 1.06i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.40 - 9.80i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (4.17 + 1.22i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-4.26 - 9.34i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-4.74 - 10.3i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.746 + 0.861i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (2.28 - 15.8i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (0.887 - 0.570i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.62 + 0.772i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (11.4 + 7.35i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.71862374183207281072585211877, −12.41169748373739656686774236902, −11.61052841562796753005674660067, −10.03423393090383360175484294996, −8.764770722350885237244156467175, −8.240380692864954449853908390542, −6.71863962404569405604738389460, −5.78604642742996106133560856568, −4.22967354087050032802936743124, −2.59219720375274437005176621041,
1.93421120601844444960567385219, 3.76564756702632490519309041234, 4.69564151398396659296988759820, 6.51439805727264983744939721239, 7.81763202049222725316530848987, 9.105113483428444834711328012109, 10.05538773687737017732067305242, 10.91212265077512553890304822600, 12.09800116622634487596177130263, 13.21051054324436200567309348887
