| L(s) = 1 | − 0.0180·2-s − 4.14·3-s − 7.99·4-s + 13.3·5-s + 0.0747·6-s + 5.69·7-s + 0.288·8-s − 9.78·9-s − 0.240·10-s − 47.5·11-s + 33.1·12-s + 76.9·13-s − 0.102·14-s − 55.3·15-s + 63.9·16-s + 7.02·17-s + 0.176·18-s − 132.·19-s − 106.·20-s − 23.6·21-s + 0.855·22-s + 93.7·23-s − 1.19·24-s + 53.2·25-s − 1.38·26-s + 152.·27-s − 45.5·28-s + ⋯ |
| L(s) = 1 | − 0.00636·2-s − 0.798·3-s − 0.999·4-s + 1.19·5-s + 0.00508·6-s + 0.307·7-s + 0.0127·8-s − 0.362·9-s − 0.00760·10-s − 1.30·11-s + 0.798·12-s + 1.64·13-s − 0.00195·14-s − 0.953·15-s + 0.999·16-s + 0.100·17-s + 0.00230·18-s − 1.59·19-s − 1.19·20-s − 0.245·21-s + 0.00829·22-s + 0.850·23-s − 0.0101·24-s + 0.425·25-s − 0.0104·26-s + 1.08·27-s − 0.307·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.0180T + 8T^{2} \) |
| 3 | \( 1 + 4.14T + 27T^{2} \) |
| 5 | \( 1 - 13.3T + 125T^{2} \) |
| 7 | \( 1 - 5.69T + 343T^{2} \) |
| 11 | \( 1 + 47.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.02T + 4.91e3T^{2} \) |
| 19 | \( 1 + 132.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 169.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 268.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 157.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 447.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 168.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 646.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 61.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 336.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 664.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 242.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−8.692421960869370621354617767449, −7.919788384960842971710899157236, −6.57209515873030834900436530741, −5.86325764317778341958657934616, −5.38205008749563628354952094829, −4.66190956733566980328123720218, −3.50584488563459638221070771210, −2.28168146971027412761749438073, −1.09546020270611051181816848601, 0,
1.09546020270611051181816848601, 2.28168146971027412761749438073, 3.50584488563459638221070771210, 4.66190956733566980328123720218, 5.38205008749563628354952094829, 5.86325764317778341958657934616, 6.57209515873030834900436530741, 7.919788384960842971710899157236, 8.692421960869370621354617767449
