| L(s) = 1 | + (0.394 + 1.35i)2-s − 0.246i·3-s + (−1.68 + 1.07i)4-s + (0.335 − 0.0972i)6-s − 1.50i·7-s + (−2.12 − 1.87i)8-s + 2.93·9-s + 6.25i·11-s + (0.264 + 0.416i)12-s + (2.04 − 0.593i)14-s + (1.70 − 3.61i)16-s + (1.15 + 3.99i)18-s + 8.33i·19-s − 0.371·21-s + (−8.50 + 2.46i)22-s + ⋯ |
| L(s) = 1 | + (0.278 + 0.960i)2-s − 0.142i·3-s + (−0.844 + 0.535i)4-s + (0.136 − 0.0397i)6-s − 0.569i·7-s + (−0.749 − 0.661i)8-s + 0.979·9-s + 1.88i·11-s + (0.0762 + 0.120i)12-s + (0.546 − 0.158i)14-s + (0.426 − 0.904i)16-s + (0.273 + 0.940i)18-s + 1.91i·19-s − 0.0810·21-s + (−1.81 + 0.526i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.731656 + 1.33038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.731656 + 1.33038i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.394 - 1.35i)T \) |
| 167 | \( 1 + 12.9iT \) |
| good | 3 | \( 1 + 0.246iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 1.50iT - 7T^{2} \) |
| 11 | \( 1 - 6.25iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8.33iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 8.65iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 10.0iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−10.39747862359522993540433411242, −9.972146509598177969025439813386, −8.996716598875357085429901263036, −7.81514234949845255803318350238, −7.23912585299556245571292064350, −6.63672303541079836986072654004, −5.33483478433815181109780326291, −4.44141454973687728521619748164, −3.65877088955350178974357095955, −1.66936289729828131947408814971,
0.808198360027888243187490689079, 2.44866486334040534696167452570, 3.43611295655392651087365634236, 4.52317257088825890970746280029, 5.49157559621889575030254291213, 6.39539750509654645560848426561, 7.76421733523745926562411746465, 9.028558431741489499568680313247, 9.184289279802678988212840881735, 10.46610930048674285826660491163
