L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 3.25·5-s + 0.999·8-s + (−1.62 + 2.82i)10-s − 5.62·11-s + (0.613 − 1.06i)13-s + (−0.5 + 0.866i)16-s + (−2.95 + 5.11i)17-s + (−1.32 − 2.29i)19-s + (−1.62 − 2.82i)20-s + (2.81 − 4.87i)22-s − 6.62·23-s + 5.62·25-s + (0.613 + 1.06i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.45·5-s + 0.353·8-s + (−0.515 + 0.892i)10-s − 1.69·11-s + (0.170 − 0.294i)13-s + (−0.125 + 0.216i)16-s + (−0.716 + 1.24i)17-s + (−0.303 − 0.525i)19-s + (−0.364 − 0.631i)20-s + (0.599 − 1.03i)22-s − 1.38·23-s + 1.12·25-s + (0.120 + 0.208i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2929117042\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2929117042\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.25T + 5T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 + (-0.613 + 1.06i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.95 - 5.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.32 + 2.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.613 + 1.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.95 - 5.11i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.81 + 6.60i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.29 + 9.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.43 + 12.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.24 + 3.88i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.81 - 4.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.6T + 71T^{2} \) |
| 73 | \( 1 + (5.59 - 9.69i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.68 - 2.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.87 + 6.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.48 - 7.77i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.53 + 2.65i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.429555018656485175960665437569, −8.045851007292391166990591341800, −6.99114356860536222402437595583, −6.20283937824110281478990450624, −5.63237532463094590399498698824, −5.05070673315441269684117090118, −3.87850473391208271392249612657, −2.41339308197948482830687826391, −1.87527306326502009730232128928, −0.094160489831582499729300601183,
1.59803066949590523785401569170, 2.36274910439347083200221068899, 3.07124423888081159938115425376, 4.44467472525215346631371653193, 5.25117302630386239003768098642, 5.92422159563954018824993740724, 6.85166162772227974434768575390, 7.75665769152345513146482423045, 8.503312093407194190959228028073, 9.337479540116643496818877829175