| L(s) = 1 | + (0.577 + 0.794i)2-s + (0.937 − 2.88i)4-s + (0.321 + 0.233i)5-s + (6.87 + 2.23i)7-s + (6.57 − 2.13i)8-s + 0.390i·10-s + (0.495 − 10.9i)11-s + (7.14 + 9.83i)13-s + (2.19 + 6.75i)14-s + (−4.32 − 3.14i)16-s + (−12.2 + 16.9i)17-s + (−10.5 + 3.43i)19-s + (0.976 − 0.709i)20-s + (9.02 − 5.95i)22-s + 5.92·23-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.397i)2-s + (0.234 − 0.721i)4-s + (0.0643 + 0.0467i)5-s + (0.981 + 0.319i)7-s + (0.821 − 0.266i)8-s + 0.0390i·10-s + (0.0450 − 0.998i)11-s + (0.549 + 0.756i)13-s + (0.156 + 0.482i)14-s + (−0.270 − 0.196i)16-s + (−0.722 + 0.995i)17-s + (−0.556 + 0.180i)19-s + (0.0488 − 0.0354i)20-s + (0.410 − 0.270i)22-s + 0.257·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.74866 + 0.0214181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74866 + 0.0214181i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-0.495 + 10.9i)T \) |
| good | 2 | \( 1 + (-0.577 - 0.794i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (-0.321 - 0.233i)T + (7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (-6.87 - 2.23i)T + (39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-7.14 - 9.83i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (12.2 - 16.9i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (10.5 - 3.43i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 - 5.92T + 529T^{2} \) |
| 29 | \( 1 + (23.7 + 7.71i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (48.2 - 35.0i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (1.84 - 5.67i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-49.4 + 16.0i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 17.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-17.1 - 52.6i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (76.5 - 55.6i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-7.75 + 23.8i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-19.6 + 27.0i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 94.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-66.9 - 48.6i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (44.2 + 14.3i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-35.4 - 48.7i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-88.5 + 121. i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 134.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (30.4 - 22.1i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−14.04934099070196991700984251928, −12.77856033827240320403883308620, −11.16898503722818101256676870696, −10.84827366615334829847190912653, −9.173585808790202705768841755450, −8.091160099884757864551310262648, −6.55072414634032557816569033461, −5.63282027720232978959807838099, −4.24041566188056065333864807622, −1.77342794680077379979367960715,
2.08438195820690602110323169999, 3.88815814113012301891352573084, 5.11946847460624639786969549682, 7.11277015336112789863426057674, 7.937438669881503321332253986798, 9.288947305480038421266233732371, 10.90140510832281448142038857460, 11.39863732667651231514222090862, 12.71777231740817010036513527598, 13.35072684340561101116925415671
