| L(s) = 1 | − 1.70·2-s − 3-s + 0.903·4-s + 3.92·5-s + 1.70·6-s − 2.84·7-s + 1.86·8-s + 9-s − 6.68·10-s − 1.86·11-s − 0.903·12-s + 6.59·13-s + 4.85·14-s − 3.92·15-s − 4.99·16-s + 17-s − 1.70·18-s + 5.88·19-s + 3.54·20-s + 2.84·21-s + 3.18·22-s + 8.09·23-s − 1.86·24-s + 10.3·25-s − 11.2·26-s − 27-s − 2.57·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.451·4-s + 1.75·5-s + 0.695·6-s − 1.07·7-s + 0.660·8-s + 0.333·9-s − 2.11·10-s − 0.562·11-s − 0.260·12-s + 1.83·13-s + 1.29·14-s − 1.01·15-s − 1.24·16-s + 0.242·17-s − 0.401·18-s + 1.34·19-s + 0.792·20-s + 0.621·21-s + 0.678·22-s + 1.68·23-s − 0.381·24-s + 2.07·25-s − 2.20·26-s − 0.192·27-s − 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438997537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438997537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.70T + 2T^{2} \) |
| 5 | \( 1 - 3.92T + 5T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 - 7.49T + 29T^{2} \) |
| 31 | \( 1 + 0.0109T + 31T^{2} \) |
| 37 | \( 1 + 3.18T + 37T^{2} \) |
| 41 | \( 1 - 8.13T + 41T^{2} \) |
| 43 | \( 1 - 8.10T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 - 6.72T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 9.25T + 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.954076034732381238397007640675, −6.91652280941415266945719361662, −6.68248053724359434825980051525, −5.62193877137327893210617286703, −5.56982935812201279696528083671, −4.40847150052676914106491500429, −3.21208025873700042457468375077, −2.48763997121077760491462531812, −1.19482868728462502031775570480, −0.941033073471259039084860763516,
0.941033073471259039084860763516, 1.19482868728462502031775570480, 2.48763997121077760491462531812, 3.21208025873700042457468375077, 4.40847150052676914106491500429, 5.56982935812201279696528083671, 5.62193877137327893210617286703, 6.68248053724359434825980051525, 6.91652280941415266945719361662, 7.954076034732381238397007640675
