L(s) = 1 | − 2.50·2-s + 2.01·3-s + 4.27·4-s + 3.19·5-s − 5.03·6-s + 0.790·7-s − 5.68·8-s + 1.04·9-s − 7.99·10-s − 2.85·11-s + 8.59·12-s − 2.42·13-s − 1.98·14-s + 6.41·15-s + 5.70·16-s + 2.08·17-s − 2.61·18-s + 0.238·19-s + 13.6·20-s + 1.59·21-s + 7.13·22-s + 0.00126·23-s − 11.4·24-s + 5.17·25-s + 6.06·26-s − 3.93·27-s + 3.37·28-s + ⋯ |
L(s) = 1 | − 1.77·2-s + 1.16·3-s + 2.13·4-s + 1.42·5-s − 2.05·6-s + 0.298·7-s − 2.01·8-s + 0.347·9-s − 2.52·10-s − 0.859·11-s + 2.47·12-s − 0.671·13-s − 0.529·14-s + 1.65·15-s + 1.42·16-s + 0.504·17-s − 0.615·18-s + 0.0547·19-s + 3.04·20-s + 0.347·21-s + 1.52·22-s + 0.000264·23-s − 2.33·24-s + 1.03·25-s + 1.18·26-s − 0.757·27-s + 0.638·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 6047 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 2.01T + 3T^{2} \) |
| 5 | \( 1 - 3.19T + 5T^{2} \) |
| 7 | \( 1 - 0.790T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 0.238T + 19T^{2} \) |
| 23 | \( 1 - 0.00126T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 + 9.00T + 31T^{2} \) |
| 37 | \( 1 + 0.877T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.84T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 8.15T + 79T^{2} \) |
| 83 | \( 1 - 5.00T + 83T^{2} \) |
| 89 | \( 1 + 18.2T + 89T^{2} \) |
| 97 | \( 1 - 6.52T + 97T^{2} \) |
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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
−7.86698751406218639863147351667, −7.43526718075550367685090146268, −6.67256476836332357061528651738, −5.67212096373183390767123669667, −5.16927632005411074686832603365, −3.56014404357042658192403950508, −2.69695438820202691650022348910, −2.02150824620541957741016502003, −1.61661755526275070942037211099, 0,
1.61661755526275070942037211099, 2.02150824620541957741016502003, 2.69695438820202691650022348910, 3.56014404357042658192403950508, 5.16927632005411074686832603365, 5.67212096373183390767123669667, 6.67256476836332357061528651738, 7.43526718075550367685090146268, 7.86698751406218639863147351667