| L(s) = 1 | + (−24.7 + 22.9i)2-s + (116. − 17.5i)3-s + (46.8 − 624. i)4-s + (−217. + 554. i)5-s + (−2.47e3 + 3.10e3i)6-s + (5.30e3 − 3.48e3i)7-s + (2.41e3 + 3.02e3i)8-s + (−5.59e3 + 1.72e3i)9-s + (−7.34e3 − 1.87e4i)10-s + (−1.88e4 − 5.81e3i)11-s + (−5.50e3 − 7.34e4i)12-s + (2.00e3 + 8.77e3i)13-s + (−5.11e4 + 2.08e5i)14-s + (−1.55e4 + 6.82e4i)15-s + (1.88e5 + 2.83e4i)16-s + (5.48e4 − 3.74e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.09 + 1.01i)2-s + (0.828 − 0.124i)3-s + (0.0914 − 1.22i)4-s + (−0.155 + 0.396i)5-s + (−0.779 + 0.977i)6-s + (0.835 − 0.549i)7-s + (0.208 + 0.261i)8-s + (−0.284 + 0.0876i)9-s + (−0.232 − 0.591i)10-s + (−0.388 − 0.119i)11-s + (−0.0766 − 1.02i)12-s + (0.0194 + 0.0852i)13-s + (−0.356 + 1.44i)14-s + (−0.0794 + 0.348i)15-s + (0.717 + 0.108i)16-s + (0.159 − 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.321887 + 1.06084i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.321887 + 1.06084i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.30e3 + 3.48e3i)T \) |
| good | 2 | \( 1 + (24.7 - 22.9i)T + (38.2 - 510. i)T^{2} \) |
| 3 | \( 1 + (-116. + 17.5i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (217. - 554. i)T + (-1.43e6 - 1.32e6i)T^{2} \) |
| 11 | \( 1 + (1.88e4 + 5.81e3i)T + (1.94e9 + 1.32e9i)T^{2} \) |
| 13 | \( 1 + (-2.00e3 - 8.77e3i)T + (-9.55e9 + 4.60e9i)T^{2} \) |
| 17 | \( 1 + (-5.48e4 + 3.74e4i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-5.02e5 - 8.70e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-1.25e6 - 8.58e5i)T + (6.58e11 + 1.67e12i)T^{2} \) |
| 29 | \( 1 + (5.67e6 + 2.73e6i)T + (9.04e12 + 1.13e13i)T^{2} \) |
| 31 | \( 1 + (5.78e5 - 1.00e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-9.64e5 - 1.28e7i)T + (-1.28e14 + 1.93e13i)T^{2} \) |
| 41 | \( 1 + (5.45e6 + 6.83e6i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 43 | \( 1 + (1.62e7 - 2.04e7i)T + (-1.11e14 - 4.89e14i)T^{2} \) |
| 47 | \( 1 + (-1.38e7 + 1.28e7i)T + (8.36e13 - 1.11e15i)T^{2} \) |
| 53 | \( 1 + (3.91e6 - 5.22e7i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (1.74e7 + 4.45e7i)T + (-6.35e15 + 5.89e15i)T^{2} \) |
| 61 | \( 1 + (-1.25e7 - 1.67e8i)T + (-1.15e16 + 1.74e15i)T^{2} \) |
| 67 | \( 1 + (1.18e8 - 2.05e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (9.27e7 - 4.46e7i)T + (2.85e16 - 3.58e16i)T^{2} \) |
| 73 | \( 1 + (-2.53e6 - 2.35e6i)T + (4.39e15 + 5.87e16i)T^{2} \) |
| 79 | \( 1 + (1.52e6 + 2.64e6i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.54e6 + 6.78e6i)T + (-1.68e17 - 8.11e16i)T^{2} \) |
| 89 | \( 1 + (-2.80e8 + 8.64e7i)T + (2.89e17 - 1.97e17i)T^{2} \) |
| 97 | \( 1 - 1.44e9T + 7.60e17T^{2} \) |
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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
−14.55019475366964071523197658064, −13.41819934147411926660336943849, −11.47218858670723971283987005750, −10.15927680835687032373257663641, −8.929604652466425694000043437230, −7.87016843172582785136088760873, −7.33134545677537959019829686901, −5.57081625818361121733409849035, −3.35599872084430214663125988128, −1.36561882531670557608670893096,
0.50761876306596864116920772767, 2.04226194506672879340150354224, 3.11926760662664877760975782062, 5.14981164914268946729115817435, 7.67478174467340452442508950602, 8.767448878758107305580580835835, 9.236307674235427798743650166941, 10.80115214729110553180722026769, 11.66130913815332154214574560117, 12.88396434539021979249002152096
