| L(s) = 1 | + (−103. + 178. i)5-s + (−503. − 872. i)7-s + (1.65e3 + 2.86e3i)11-s + (−6.64e3 + 1.15e4i)13-s + 1.80e4·17-s + 2.17e4·19-s + (3.24e4 − 5.61e4i)23-s + (1.77e4 + 3.07e4i)25-s + (−1.05e5 − 1.83e5i)29-s + (−4.23e4 + 7.33e4i)31-s + 2.07e5·35-s + 1.63e5·37-s + (−2.70e5 + 4.68e5i)41-s + (1.52e5 + 2.64e5i)43-s + (4.05e5 + 7.01e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.369 + 0.639i)5-s + (−0.554 − 0.960i)7-s + (0.374 + 0.648i)11-s + (−0.838 + 1.45i)13-s + 0.888·17-s + 0.726·19-s + (0.555 − 0.962i)23-s + (0.227 + 0.393i)25-s + (−0.804 − 1.39i)29-s + (−0.255 + 0.442i)31-s + 0.819·35-s + 0.530·37-s + (−0.613 + 1.06i)41-s + (0.292 + 0.507i)43-s + (0.569 + 0.985i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2650702766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2650702766\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (103. - 178. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (503. + 872. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.65e3 - 2.86e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (6.64e3 - 1.15e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.17e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.24e4 + 5.61e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.05e5 + 1.83e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (4.23e4 - 7.33e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.63e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.70e5 - 4.68e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.52e5 - 2.64e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-4.05e5 - 7.01e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.40e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (6.98e4 - 1.20e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.45e5 + 2.52e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-8.08e5 + 1.39e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 9.62e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.11e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.76e6 + 6.51e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.74e6 - 3.01e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 9.64e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (4.91e5 + 8.52e5i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−10.91719709099265262653370275788, −9.836337482182259244847643817807, −9.363512355969480625428990957427, −7.73220967040862063057733758467, −7.11897489631107004609757071608, −6.36752676334236370111948057167, −4.75592723296385017019387061608, −3.87775162416204743729652347546, −2.78205952430922222837775047946, −1.34489698515470762800109911068,
0.06365289208002837082959602483, 1.14102178828525723933965280377, 2.78231948717338338927258047040, 3.60555064732556801928626272025, 5.29452950969605671642180472780, 5.61908945890789231461971953592, 7.15171193388493686327612795232, 8.088948384782378665877140659938, 9.006236976686337419284987591536, 9.743632253765345871993219469183
