| L(s) = 1 | + (−1.40 − 0.118i)2-s + (0.991 − 0.130i)3-s + (1.97 + 0.333i)4-s + (−1.89 − 0.249i)5-s + (−1.41 + 0.0664i)6-s + (1.39 − 2.24i)7-s + (−2.73 − 0.704i)8-s + (0.965 − 0.258i)9-s + (2.64 + 0.576i)10-s + (−4.16 − 3.19i)11-s + (1.99 + 0.0737i)12-s + (−0.437 + 1.05i)13-s + (−2.23 + 2.99i)14-s − 1.91·15-s + (3.77 + 1.31i)16-s + (−0.928 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 − 0.0837i)2-s + (0.572 − 0.0753i)3-s + (0.985 + 0.166i)4-s + (−0.848 − 0.111i)5-s + (−0.576 + 0.0271i)6-s + (0.529 − 0.848i)7-s + (−0.968 − 0.248i)8-s + (0.321 − 0.0862i)9-s + (0.836 + 0.182i)10-s + (−1.25 − 0.964i)11-s + (0.576 + 0.0212i)12-s + (−0.121 + 0.292i)13-s + (−0.598 + 0.801i)14-s − 0.494·15-s + (0.944 + 0.329i)16-s + (−0.225 − 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0866240 - 0.425550i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0866240 - 0.425550i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 + 0.118i)T \) |
| 3 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 + (-1.39 + 2.24i)T \) |
| good | 5 | \( 1 + (1.89 + 0.249i)T + (4.82 + 1.29i)T^{2} \) |
| 11 | \( 1 + (4.16 + 3.19i)T + (2.84 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.437 - 1.05i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (0.928 + 1.60i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.18 - 3.98i)T + (4.91 - 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.431 - 1.60i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.11 - 2.70i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (4.29 + 7.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.459 - 3.48i)T + (-35.7 - 9.57i)T^{2} \) |
| 41 | \( 1 + (-5.87 + 5.87i)T - 41iT^{2} \) |
| 43 | \( 1 + (1.64 - 0.682i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (0.0886 + 0.0511i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.95 - 3.85i)T + (-13.7 - 51.1i)T^{2} \) |
| 59 | \( 1 + (2.49 + 1.91i)T + (15.2 + 56.9i)T^{2} \) |
| 61 | \( 1 + (-5.82 + 4.46i)T + (15.7 - 58.9i)T^{2} \) |
| 67 | \( 1 + (1.32 + 10.0i)T + (-64.7 + 17.3i)T^{2} \) |
| 71 | \( 1 + (-6.41 - 6.41i)T + 71iT^{2} \) |
| 73 | \( 1 + (15.8 + 4.24i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.92 + 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.78 + 2.81i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (13.9 - 3.73i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 7.23iT - 97T^{2} \) |
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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
−10.18260230222724878208673022099, −9.120935701103035331788633248288, −8.149371913700865423005137188891, −7.87853141019495286229634990728, −7.06368377090645566847839204458, −5.78637619381021251135045890887, −4.27051884422726890237395428327, −3.28988232826471598416028006289, −1.96335503151121428655048347141, −0.27486740757903683105409752679,
2.03196050428185069231321451285, 2.84539282382127576375730140191, 4.40565782865387956141440512522, 5.53153672588463857946146761336, 6.88127278548971636051801230142, 7.66675124752587047532014893233, 8.311207426269859831095582906567, 8.940537055499733632227057639731, 9.960966162103045108313421215359, 10.77791629108825912916756472108
