| L(s) = 1 | + (−20.9 − 36.2i)2-s + (0.116 − 0.201i)3-s + (−617. + 1.06e3i)4-s + (−895. − 1.55e3i)5-s − 9.71·6-s + 3.02e4·8-s + (9.84e3 + 1.70e4i)9-s + (−3.74e4 + 6.48e4i)10-s + (−8.70e3 + 1.50e4i)11-s + (143. + 248. i)12-s + 1.22e5·13-s − 416.·15-s + (−3.15e5 − 5.46e5i)16-s + (1.65e5 − 2.87e5i)17-s + (4.11e5 − 7.12e5i)18-s + (3.80e5 + 6.59e5i)19-s + ⋯ |
| L(s) = 1 | + (−0.923 − 1.59i)2-s + (0.000828 − 0.00143i)3-s + (−1.20 + 2.08i)4-s + (−0.641 − 1.11i)5-s − 0.00305·6-s + 2.61·8-s + (0.499 + 0.866i)9-s + (−1.18 + 2.05i)10-s + (−0.179 + 0.310i)11-s + (0.00199 + 0.00346i)12-s + 1.18·13-s − 0.00212·15-s + (−1.20 − 2.08i)16-s + (0.481 − 0.834i)17-s + (0.923 − 1.59i)18-s + (0.670 + 1.16i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.371472 - 0.886393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371472 - 0.886393i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (20.9 + 36.2i)T + (-256 + 443. i)T^{2} \) |
| 3 | \( 1 + (-0.116 + 0.201i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (895. + 1.55e3i)T + (-9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (8.70e3 - 1.50e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.22e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.65e5 + 2.87e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-3.80e5 - 6.59e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (6.16e5 + 1.06e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 - 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + (2.69e6 - 4.65e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.51e6 - 2.62e6i)T + (-6.49e13 + 1.12e14i)T^{2} \) |
| 41 | \( 1 - 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.01e7 - 1.76e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.98e7 + 5.17e7i)T + (-1.64e15 - 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-3.01e7 + 5.22e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (4.72e6 + 8.17e6i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.09e8 + 1.89e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.27e8 + 3.93e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.25e7 + 3.90e7i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 - 3.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-3.25e8 - 5.64e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.42e9T + 7.60e17T^{2} \) |
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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
−12.68519206130449556026471272820, −11.97870502424649407531506363316, −10.81845128207365374690169401414, −9.774676599739700254116862226038, −8.538992157631286564037052315876, −7.78942102488791635971591968838, −4.80867480813723941749194793872, −3.54946695917504672057758187073, −1.75325554660391719202563721015, −0.68489388098531574280040412738,
0.867769990826638127021400170041, 3.73281845605825054402391673658, 5.81485452841041770446441662236, 6.82532621411976635907661701722, 7.73054781744083272729042788624, 8.956890368212126983654270908827, 10.17683748527626738302813761488, 11.35985136051092444582376762917, 13.44115074262180022907967025956, 14.66117562217002747269118107774
