| L(s) = 1 | + (−1.12 + 1.65i)2-s + (−1.70 − 2.47i)3-s + (−1.47 − 3.71i)4-s + (−0.0706 + 0.0140i)5-s + (5.99 − 0.0397i)6-s + (−2.03 − 4.90i)7-s + (7.80 + 1.74i)8-s + (−3.20 + 8.40i)9-s + (0.0561 − 0.132i)10-s + (−1.40 + 2.09i)11-s + (−6.67 + 9.97i)12-s + (−1.41 + 7.12i)13-s + (10.3 + 2.14i)14-s + (0.154 + 0.150i)15-s + (−11.6 + 10.9i)16-s + (−11.2 + 11.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.561 + 0.827i)2-s + (−0.567 − 0.823i)3-s + (−0.368 − 0.929i)4-s + (−0.0141 + 0.00280i)5-s + (0.999 − 0.00661i)6-s + (−0.290 − 0.700i)7-s + (0.976 + 0.217i)8-s + (−0.356 + 0.934i)9-s + (0.00561 − 0.0132i)10-s + (−0.127 + 0.190i)11-s + (−0.556 + 0.830i)12-s + (−0.108 + 0.547i)13-s + (0.742 + 0.153i)14-s + (0.0103 + 0.0100i)15-s + (−0.728 + 0.685i)16-s + (−0.661 + 0.661i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0256494 + 0.155914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0256494 + 0.155914i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 - 1.65i)T \) |
| 3 | \( 1 + (1.70 + 2.47i)T \) |
| good | 5 | \( 1 + (0.0706 - 0.0140i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (2.03 + 4.90i)T + (-34.6 + 34.6i)T^{2} \) |
| 11 | \( 1 + (1.40 - 2.09i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (1.41 - 7.12i)T + (-156. - 64.6i)T^{2} \) |
| 17 | \( 1 + (11.2 - 11.2i)T - 289iT^{2} \) |
| 19 | \( 1 + (11.8 + 2.36i)T + (333. + 138. i)T^{2} \) |
| 23 | \( 1 + (8.45 - 20.4i)T + (-374. - 374. i)T^{2} \) |
| 29 | \( 1 + (-4.70 - 7.04i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 + 14.8iT - 961T^{2} \) |
| 37 | \( 1 + (17.6 - 3.50i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (14.3 - 34.7i)T + (-1.18e3 - 1.18e3i)T^{2} \) |
| 43 | \( 1 + (-20.6 + 30.8i)T + (-707. - 1.70e3i)T^{2} \) |
| 47 | \( 1 + (57.2 - 57.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (19.2 - 28.8i)T + (-1.07e3 - 2.59e3i)T^{2} \) |
| 59 | \( 1 + (-94.0 + 18.7i)T + (3.21e3 - 1.33e3i)T^{2} \) |
| 61 | \( 1 + (-3.43 - 5.14i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + (15.5 + 23.3i)T + (-1.71e3 + 4.14e3i)T^{2} \) |
| 71 | \( 1 + (98.6 - 40.8i)T + (3.56e3 - 3.56e3i)T^{2} \) |
| 73 | \( 1 + (31.4 - 75.8i)T + (-3.76e3 - 3.76e3i)T^{2} \) |
| 79 | \( 1 + (-55.2 + 55.2i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + (-20.1 + 101. i)T + (-6.36e3 - 2.63e3i)T^{2} \) |
| 89 | \( 1 + (39.4 + 95.3i)T + (-5.60e3 + 5.60e3i)T^{2} \) |
| 97 | \( 1 + 95.5iT - 9.40e3T^{2} \) |
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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
−13.03056689965518024122524035157, −11.62969601711487714793838530070, −10.71223021113026110898455805466, −9.741841983647953311081640759435, −8.461642952259167185127189415595, −7.47077918698564462510460666808, −6.69224419193825677297455014192, −5.75752882317742827717200432110, −4.36476187856762036826846698511, −1.70983117084666966280342204064,
0.11544885872665480241829956012, 2.57297816140030233847255363755, 3.95041748458960543220854796798, 5.20587669963764285080921883234, 6.58039190273641000525031074821, 8.241560626690235311578742076433, 9.096562347670308664700971173229, 10.03866993925468596710414030139, 10.76832586011740531561619775532, 11.77851764149536081882516823643
