L(s) = 1 | + (5.85 − 5.85i)2-s + (−33.0 − 33.0i)3-s + 187. i·4-s − 387.·6-s + (−1.10e3 + 1.10e3i)7-s + (2.59e3 + 2.59e3i)8-s + 2.18e3i·9-s + 2.03e4·11-s + (6.19e3 − 6.19e3i)12-s + (−3.80e4 − 3.80e4i)13-s + 1.29e4i·14-s − 1.75e4·16-s + (−3.50e4 + 3.50e4i)17-s + (1.28e4 + 1.28e4i)18-s − 3.37e4i·19-s + ⋯ |
L(s) = 1 | + (0.366 − 0.366i)2-s + (−0.408 − 0.408i)3-s + 0.731i·4-s − 0.298·6-s + (−0.460 + 0.460i)7-s + (0.634 + 0.634i)8-s + 0.333i·9-s + 1.38·11-s + (0.298 − 0.298i)12-s + (−1.33 − 1.33i)13-s + 0.337i·14-s − 0.267·16-s + (−0.420 + 0.420i)17-s + (0.122 + 0.122i)18-s − 0.259i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0169060 - 0.257453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0169060 - 0.257453i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (33.0 + 33.0i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-5.85 + 5.85i)T - 256iT^{2} \) |
| 7 | \( 1 + (1.10e3 - 1.10e3i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 - 2.03e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (3.80e4 + 3.80e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (3.50e4 - 3.50e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + 3.37e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (1.80e5 + 1.80e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + 1.26e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 6.06e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-3.25e4 + 3.25e4i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + 4.66e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (2.52e6 + 2.52e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (-2.01e6 + 2.01e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + (6.90e6 + 6.90e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 - 3.09e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.67e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-5.43e6 + 5.43e6i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 - 4.05e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.17e6 - 1.17e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 5.57e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (3.11e7 + 3.11e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + 5.37e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-9.15e7 + 9.15e7i)T - 7.83e15iT^{2} \) |
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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
−12.33801944384179787325656773652, −11.67676496360063841363196264435, −10.30255677291268096214839098346, −8.850675353050920453555639786741, −7.59877503621900233605740372126, −6.39115449999681222698120131363, −4.88890956264858160132634108630, −3.40392555579725696168715626349, −2.06331990813039703796210580145, −0.07168220306392732033553461744,
1.59459868763910013108306565049, 3.92661440349715693529250722212, 4.91663153662969126540286096999, 6.37126626714963362673547922041, 7.08147541399129160215365960177, 9.322811915948726154205269628070, 9.868647833023726537074973639205, 11.23724436434418873511985756151, 12.20971680301014227728797185665, 13.77371254868524450822052198721