L(s) = 1 | + (−0.265 + 1.38i)2-s + 0.0612i·3-s + (−1.85 − 0.737i)4-s − i·5-s + (−0.0850 − 0.0162i)6-s − 0.810·7-s + (1.51 − 2.38i)8-s + 2.99·9-s + (1.38 + 0.265i)10-s − 2.10·11-s + (0.0451 − 0.113i)12-s + 2.93·13-s + (0.215 − 1.12i)14-s + 0.0612·15-s + (2.91 + 2.74i)16-s − 4.65i·17-s + ⋯ |
L(s) = 1 | + (−0.187 + 0.982i)2-s + 0.0353i·3-s + (−0.929 − 0.368i)4-s − 0.447i·5-s + (−0.0347 − 0.00663i)6-s − 0.306·7-s + (0.536 − 0.843i)8-s + 0.998·9-s + (0.439 + 0.0838i)10-s − 0.633·11-s + (0.0130 − 0.0328i)12-s + 0.814·13-s + (0.0574 − 0.301i)14-s + 0.0158·15-s + (0.728 + 0.685i)16-s − 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17049 + 0.376503i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17049 + 0.376503i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.265 - 1.38i)T \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-4.03 - 2.59i)T \) |
good | 3 | \( 1 - 0.0612iT - 3T^{2} \) |
| 7 | \( 1 + 0.810T + 7T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 4.65iT - 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 - 2.58iT - 31T^{2} \) |
| 37 | \( 1 + 2.99iT - 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 1.89iT - 47T^{2} \) |
| 53 | \( 1 + 8.12iT - 53T^{2} \) |
| 59 | \( 1 + 13.1iT - 59T^{2} \) |
| 61 | \( 1 - 13.7iT - 61T^{2} \) |
| 67 | \( 1 - 1.42T + 67T^{2} \) |
| 71 | \( 1 + 2.26iT - 71T^{2} \) |
| 73 | \( 1 + 4.13T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 8.73T + 83T^{2} \) |
| 89 | \( 1 - 5.82iT - 89T^{2} \) |
| 97 | \( 1 - 2.73iT - 97T^{2} \) |
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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
−10.97852896851565101860747736778, −9.847098420096284552363843443167, −9.390527236099553915592277144742, −8.309580702622867806992367748345, −7.42005637937314267802603369159, −6.67691828985568157442388606891, −5.41492611648461168581341565613, −4.72881257974535400125454354572, −3.38091105349331994870739327899, −1.06726062452361544607458848497,
1.30552971756283034367686624868, 2.83713156862426828571034575997, 3.82950017695992429698121672995, 4.96877993887305450983169809771, 6.30669604253359202587465318801, 7.49898395359290323512437948048, 8.365966652616469907780985868809, 9.455300364143450914034286261359, 10.24381445424555626802090763894, 10.80598195217232005446313561769