| 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
−10.77791629108825912916756472108, −9.960966162103045108313421215359, −8.940537055499733632227057639731, −8.311207426269859831095582906567, −7.66675124752587047532014893233, −6.88127278548971636051801230142, −5.53153672588463857946146761336, −4.40565782865387956141440512522, −2.84539282382127576375730140191, −2.03196050428185069231321451285,
0.27486740757903683105409752679, 1.96335503151121428655048347141, 3.28988232826471598416028006289, 4.27051884422726890237395428327, 5.78637619381021251135045890887, 7.06368377090645566847839204458, 7.87853141019495286229634990728, 8.149371913700865423005137188891, 9.120935701103035331788633248288, 10.18260230222724878208673022099
