| L(s) = 1 | − 0.832·2-s + 3-s − 1.30·4-s − 5-s − 0.832·6-s + 2.56·7-s + 2.75·8-s + 9-s + 0.832·10-s − 1.27·11-s − 1.30·12-s + 6.61·13-s − 2.13·14-s − 15-s + 0.321·16-s + 5.71·17-s − 0.832·18-s + 5.39·19-s + 1.30·20-s + 2.56·21-s + 1.06·22-s + 4.92·23-s + 2.75·24-s + 25-s − 5.50·26-s + 27-s − 3.34·28-s + ⋯ |
| L(s) = 1 | − 0.588·2-s + 0.577·3-s − 0.653·4-s − 0.447·5-s − 0.339·6-s + 0.968·7-s + 0.973·8-s + 0.333·9-s + 0.263·10-s − 0.384·11-s − 0.377·12-s + 1.83·13-s − 0.570·14-s − 0.258·15-s + 0.0803·16-s + 1.38·17-s − 0.196·18-s + 1.23·19-s + 0.292·20-s + 0.559·21-s + 0.226·22-s + 1.02·23-s + 0.561·24-s + 0.200·25-s − 1.08·26-s + 0.192·27-s − 0.633·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.138617396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.138617396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 0.832T + 2T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 1.27T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 - 5.71T + 17T^{2} \) |
| 19 | \( 1 - 5.39T + 19T^{2} \) |
| 23 | \( 1 - 4.92T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 + 0.432T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 + 5.79T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 7.75T + 79T^{2} \) |
| 83 | \( 1 + 6.51T + 83T^{2} \) |
| 89 | \( 1 + 6.90T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.076569007291668782557128334826, −7.81021181259303227820000541615, −6.99397571545439756309849460655, −5.82085136044825347001188808697, −5.12586259625048783116302086100, −4.39743178486668469584439667474, −3.60264229563342028871305303996, −2.92276391253753983017603198236, −1.35194603324554724899053174462, −1.02644066915683968456606568273,
1.02644066915683968456606568273, 1.35194603324554724899053174462, 2.92276391253753983017603198236, 3.60264229563342028871305303996, 4.39743178486668469584439667474, 5.12586259625048783116302086100, 5.82085136044825347001188808697, 6.99397571545439756309849460655, 7.81021181259303227820000541615, 8.076569007291668782557128334826
