| L(s) = 1 | − 5.11·2-s + 7.70·3-s + 18.1·4-s + 4.71·5-s − 39.4·6-s − 22.4·7-s − 51.9·8-s + 32.3·9-s − 24.1·10-s + 139.·12-s − 13·13-s + 115.·14-s + 36.3·15-s + 120.·16-s − 21.2·17-s − 165.·18-s + 119.·19-s + 85.6·20-s − 173.·21-s − 185.·23-s − 400.·24-s − 102.·25-s + 66.4·26-s + 41.5·27-s − 408.·28-s + 25.2·29-s − 186.·30-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.48·3-s + 2.26·4-s + 0.422·5-s − 2.68·6-s − 1.21·7-s − 2.29·8-s + 1.19·9-s − 0.763·10-s + 3.36·12-s − 0.277·13-s + 2.19·14-s + 0.626·15-s + 1.88·16-s − 0.303·17-s − 2.16·18-s + 1.44·19-s + 0.957·20-s − 1.80·21-s − 1.68·23-s − 3.40·24-s − 0.821·25-s + 0.501·26-s + 0.296·27-s − 2.75·28-s + 0.161·29-s − 1.13·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 5.11T + 8T^{2} \) |
| 3 | \( 1 - 7.70T + 27T^{2} \) |
| 5 | \( 1 - 4.71T + 125T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 17 | \( 1 + 21.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 185.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 25.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 384.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 294.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 309.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 578.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 797.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 755.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 565.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 31.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 935.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 217.T + 9.12e5T^{2} \) |
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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
−8.857933796989530060129589866675, −7.932104412078907489613694643989, −7.59032685000135587985351175178, −6.60147830307913104322001983544, −5.87666895711629529001287754505, −3.99861573740263575346517544651, −2.88742711410990363214331838363, −2.42240954251196078550132564908, −1.30965898869211220203185761904, 0,
1.30965898869211220203185761904, 2.42240954251196078550132564908, 2.88742711410990363214331838363, 3.99861573740263575346517544651, 5.87666895711629529001287754505, 6.60147830307913104322001983544, 7.59032685000135587985351175178, 7.932104412078907489613694643989, 8.857933796989530060129589866675
