L(s) = 1 | + (0.119 − 0.207i)5-s + (−2.56 − 4.43i)11-s + (2.44 + 4.23i)13-s + (1.85 − 3.20i)17-s + (−1.83 − 3.16i)19-s + (3.71 − 6.42i)23-s + (2.47 + 4.28i)25-s + (1.73 − 3.00i)29-s − 0.717·31-s + (−2.30 − 3.98i)37-s + (2.80 + 4.85i)41-s + (−6.24 + 10.8i)43-s − 4.33·47-s + (−0.471 + 0.816i)53-s − 1.22·55-s + ⋯ |
L(s) = 1 | + (0.0534 − 0.0926i)5-s + (−0.772 − 1.33i)11-s + (0.677 + 1.17i)13-s + (0.449 − 0.777i)17-s + (−0.419 − 0.727i)19-s + (0.773 − 1.34i)23-s + (0.494 + 0.856i)25-s + (0.321 − 0.557i)29-s − 0.128·31-s + (−0.378 − 0.655i)37-s + (0.437 + 0.757i)41-s + (−0.952 + 1.64i)43-s − 0.633·47-s + (−0.0647 + 0.112i)53-s − 0.165·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.321161441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321161441\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.119 + 0.207i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.56 + 4.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.85 + 3.20i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.83 + 3.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.71 + 6.42i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 3.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.717T + 31T^{2} \) |
| 37 | \( 1 + (2.30 + 3.98i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.24 - 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 + (0.471 - 0.816i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 7.57T + 59T^{2} \) |
| 61 | \( 1 - 5.50T + 61T^{2} \) |
| 67 | \( 1 + 0.660T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + (-1.83 + 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6.22T + 79T^{2} \) |
| 83 | \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.74 + 6.48i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.060421972622498903873145543897, −7.21595999525090462675414498828, −6.47229063519674184908321428194, −5.91075492858531954760121156373, −4.94207074279321787206623360668, −4.42855972226345088081663119672, −3.22754255786117337470528362751, −2.75384511877871989359657800876, −1.46720508731898208279483342576, −0.36402150701682297550294853308,
1.22775864711823558420369696823, 2.15763229757690037494647571238, 3.19236478086197735760503865903, 3.84595081333773435549302617949, 4.94142551053796616936228273513, 5.42595909411358617435961802332, 6.25040857155314782685627344241, 7.07757268615536608544062975123, 7.73271703214368777837438446452, 8.318893882530040048047997825459