| L(s) = 1 | + (0.929 − 1.06i)2-s + (−0.831 − 0.555i)3-s + (−0.272 − 1.98i)4-s + (−4.03 − 0.802i)5-s + (−1.36 + 0.369i)6-s + (−0.0612 + 0.147i)7-s + (−2.36 − 1.55i)8-s + (0.382 + 0.923i)9-s + (−4.60 + 3.55i)10-s + (−0.294 − 0.440i)11-s + (−0.874 + 1.79i)12-s + (4.43 − 0.882i)13-s + (0.100 + 0.202i)14-s + (2.90 + 2.90i)15-s + (−3.85 + 1.07i)16-s + (4.35 − 4.35i)17-s + ⋯ |
| L(s) = 1 | + (0.657 − 0.753i)2-s + (−0.480 − 0.320i)3-s + (−0.136 − 0.990i)4-s + (−1.80 − 0.358i)5-s + (−0.557 + 0.150i)6-s + (−0.0231 + 0.0558i)7-s + (−0.836 − 0.548i)8-s + (0.127 + 0.307i)9-s + (−1.45 + 1.12i)10-s + (−0.0887 − 0.132i)11-s + (−0.252 + 0.519i)12-s + (1.22 − 0.244i)13-s + (0.0268 + 0.0541i)14-s + (0.750 + 0.750i)15-s + (−0.962 + 0.269i)16-s + (1.05 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.138781 - 0.884817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.138781 - 0.884817i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.929 + 1.06i)T \) |
| 3 | \( 1 + (0.831 + 0.555i)T \) |
| good | 5 | \( 1 + (4.03 + 0.802i)T + (4.61 + 1.91i)T^{2} \) |
| 7 | \( 1 + (0.0612 - 0.147i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.294 + 0.440i)T + (-4.20 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-4.43 + 0.882i)T + (12.0 - 4.97i)T^{2} \) |
| 17 | \( 1 + (-4.35 + 4.35i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.57 + 7.89i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (4.54 - 1.88i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (0.558 - 0.836i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 - 0.971iT - 31T^{2} \) |
| 37 | \( 1 + (0.153 - 0.771i)T + (-34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-4.03 + 1.67i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (5.11 - 3.41i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (0.409 - 0.409i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.55 + 2.32i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (-7.20 - 1.43i)T + (54.5 + 22.5i)T^{2} \) |
| 61 | \( 1 + (-3.19 - 2.13i)T + (23.3 + 56.3i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 6.82i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-3.81 + 9.19i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (5.05 + 12.2i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (1.43 + 1.43i)T + 79iT^{2} \) |
| 83 | \( 1 + (1.31 + 6.61i)T + (-76.6 + 31.7i)T^{2} \) |
| 89 | \( 1 + (-8.40 - 3.48i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 1.21iT - 97T^{2} \) |
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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
−11.90942852413386835470415164377, −11.45083812727605573653797960421, −10.67274001332134398131400594500, −9.135532456770069178328796381927, −7.991069670871314457305944299911, −6.82644041720560848231709795880, −5.38272882693787701307854081434, −4.30559350235340123391323743027, −3.18163060387753567298564692802, −0.71943580807416297249437330555,
3.76331539759806173597948120935, 3.95823692989392795519137005021, 5.67003741857315873236526815870, 6.68234134950204259572278572753, 7.962832462375286623331742255090, 8.353974540203542364103823108735, 10.27246343900776819617440214611, 11.31811324502225586410030473367, 12.09927745253304258586294953805, 12.73552358086306132331203997258
