| L(s) = 1 | + (−0.707 + 1.22i)2-s + (1.09 + 2.79i)3-s + (−0.999 − 1.73i)4-s + 8.26·5-s + (−4.19 − 0.638i)6-s + (−3.50 + 2.02i)7-s + 2.82·8-s + (−6.61 + 6.10i)9-s + (−5.84 + 10.1i)10-s + (−4.05 + 7.02i)11-s + (3.74 − 4.68i)12-s + (−8.22 − 10.0i)13-s − 5.71i·14-s + (9.02 + 23.0i)15-s + (−2.00 + 3.46i)16-s + (19.4 − 11.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.364 + 0.931i)3-s + (−0.249 − 0.433i)4-s + 1.65·5-s + (−0.699 − 0.106i)6-s + (−0.500 + 0.288i)7-s + 0.353·8-s + (−0.734 + 0.678i)9-s + (−0.584 + 1.01i)10-s + (−0.368 + 0.638i)11-s + (0.312 − 0.390i)12-s + (−0.632 − 0.774i)13-s − 0.408i·14-s + (0.601 + 1.53i)15-s + (−0.125 + 0.216i)16-s + (1.14 − 0.661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0232 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0232 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.930689 + 0.952617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930689 + 0.952617i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (-1.09 - 2.79i)T \) |
| 13 | \( 1 + (8.22 + 10.0i)T \) |
| good | 5 | \( 1 - 8.26T + 25T^{2} \) |
| 7 | \( 1 + (3.50 - 2.02i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.05 - 7.02i)T + (-60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (-19.4 + 11.2i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.90 + 5.14i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (7.58 + 4.38i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-19.0 - 10.9i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 40.2iT - 961T^{2} \) |
| 37 | \( 1 + (42.8 + 24.7i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-31.3 + 54.2i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-33.3 - 57.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 27.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 30.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-24.1 - 41.7i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (30.1 + 52.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (79.9 + 46.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-13.9 - 24.1i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + 48.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 70.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 119.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (54.1 - 93.7i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (56.5 - 32.6i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.50045534154594499011936280437, −13.79965724393300544560414398288, −12.56688453527700640851755479957, −10.51760221970650350318037644204, −9.764445696206292087261932943758, −9.233476784337835560920295839663, −7.62577431788343587124118530100, −5.90594541943657394835316839187, −5.07178258077911272434681149479, −2.65518133565501059572673280623,
1.55788259930896158710613620049, 3.02559863149556886406104635127, 5.66420641562990395665126872953, 6.87672237092768980253814531651, 8.381246679030944135795596916176, 9.567252063233021402329803779912, 10.33038412687373932600180377368, 11.96033423711454295647781487949, 12.90992741866044737631790200796, 13.80340919807666492816718637789
