| L(s) = 1 | + (−0.760 + 1.19i)2-s + (−1.44 + 1.87i)3-s + (−0.842 − 1.81i)4-s + (−2.75 + 2.11i)5-s + (−1.14 − 3.14i)6-s + (−1.14 − 2.38i)7-s + (2.80 + 0.376i)8-s + (−0.673 − 2.51i)9-s + (−0.424 − 4.89i)10-s + (2.68 − 0.353i)11-s + (4.61 + 1.03i)12-s + (1.84 − 0.762i)13-s + (3.71 + 0.450i)14-s − 8.23i·15-s + (−2.58 + 3.05i)16-s + (−6.86 + 3.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.537 + 0.842i)2-s + (−0.831 + 1.08i)3-s + (−0.421 − 0.907i)4-s + (−1.23 + 0.946i)5-s + (−0.466 − 1.28i)6-s + (−0.432 − 0.901i)7-s + (0.991 + 0.132i)8-s + (−0.224 − 0.837i)9-s + (−0.134 − 1.54i)10-s + (0.808 − 0.106i)11-s + (1.33 + 0.297i)12-s + (0.510 − 0.211i)13-s + (0.992 + 0.120i)14-s − 2.12i·15-s + (−0.645 + 0.763i)16-s + (−1.66 + 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0265035 - 0.0211645i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0265035 - 0.0211645i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.760 - 1.19i)T \) |
| 7 | \( 1 + (1.14 + 2.38i)T \) |
| good | 3 | \( 1 + (1.44 - 1.87i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (2.75 - 2.11i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-2.68 + 0.353i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 0.762i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (6.86 - 3.96i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.226 + 1.71i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (1.20 + 4.48i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.0946 - 0.228i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (4.72 + 8.18i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.74 - 2.87i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-0.823 + 0.823i)T - 41iT^{2} \) |
| 43 | \( 1 + (2.05 - 4.95i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-0.844 - 0.487i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.52 - 0.990i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.187 - 1.42i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (5.51 + 0.726i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (7.06 - 9.21i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (0.823 + 0.823i)T + 71iT^{2} \) |
| 73 | \( 1 + (-13.8 - 3.71i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.377 + 0.218i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (8.69 - 3.60i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.94 + 1.86i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 9.00T + 97T^{2} \) |
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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
−13.12021286039679121338819837639, −11.40071030141796727711028405712, −10.92006625931103672776772780238, −10.31055312593267878495542908623, −9.152486892702323039290414352897, −7.966249867483070416087030513145, −6.81496428588748003815594255315, −6.19260757077659577481342681916, −4.43251235369532650792577931579, −3.88693379664477069034288943305,
0.03792322097286779950169232852, 1.65499173764098549855411914623, 3.66224785266056936060246063596, 5.01594269883523523351171538720, 6.59546532213044111194054333920, 7.55169036320711571298057421335, 8.746471170050585795531813242233, 9.229556354749216474007693032802, 11.06558042764507189686024713837, 11.70423133787792116718803240858
