| L(s) = 1 | − 2.30·2-s + 3-s + 3.30·4-s − 5-s − 2.30·6-s − 7-s − 3.00·8-s − 2·9-s + 2.30·10-s + 1.60·11-s + 3.30·12-s + 2.30·14-s − 15-s + 0.302·16-s + 7.60·17-s + 4.60·18-s − 5.60·19-s − 3.30·20-s − 21-s − 3.69·22-s − 3·23-s − 3.00·24-s + 25-s − 5·27-s − 3.30·28-s − 6.21·29-s + 2.30·30-s + ⋯ |
| L(s) = 1 | − 1.62·2-s + 0.577·3-s + 1.65·4-s − 0.447·5-s − 0.940·6-s − 0.377·7-s − 1.06·8-s − 0.666·9-s + 0.728·10-s + 0.484·11-s + 0.953·12-s + 0.615·14-s − 0.258·15-s + 0.0756·16-s + 1.84·17-s + 1.08·18-s − 1.28·19-s − 0.738·20-s − 0.218·21-s − 0.788·22-s − 0.625·23-s − 0.612·24-s + 0.200·25-s − 0.962·27-s − 0.624·28-s − 1.15·29-s + 0.420·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 1.60T + 11T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 3.60T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 4.81T + 71T^{2} \) |
| 73 | \( 1 + 0.788T + 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 + 9.21T + 83T^{2} \) |
| 89 | \( 1 + 6.21T + 89T^{2} \) |
| 97 | \( 1 + 8.39T + 97T^{2} \) |
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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
−9.514784846335225045878441101699, −9.010127535818401603241780099655, −8.065298283972116118871095156377, −7.74204671857400998153510235072, −6.64970196823404938467808822875, −5.67871870518509227632034502953, −3.96830589901234715636627474403, −2.92772445542842670822879438575, −1.64745686230389725594731403662, 0,
1.64745686230389725594731403662, 2.92772445542842670822879438575, 3.96830589901234715636627474403, 5.67871870518509227632034502953, 6.64970196823404938467808822875, 7.74204671857400998153510235072, 8.065298283972116118871095156377, 9.010127535818401603241780099655, 9.514784846335225045878441101699
