L(s) = 1 | + (4.21 − 2.43i)2-s + (7.83 − 13.5i)4-s + (6.38 + 11.0i)5-s − 37.3i·8-s + (53.7 + 31.0i)10-s + (46.8 + 27.0i)11-s − 8.85i·13-s + (−28.1 − 48.7i)16-s + (−34.4 + 59.6i)17-s + (141. − 81.9i)19-s + 200.·20-s + 263.·22-s + (−81.3 + 46.9i)23-s + (−18.9 + 32.8i)25-s + (−21.5 − 37.3i)26-s + ⋯ |
L(s) = 1 | + (1.48 − 0.860i)2-s + (0.979 − 1.69i)4-s + (0.570 + 0.988i)5-s − 1.64i·8-s + (1.70 + 0.981i)10-s + (1.28 + 0.741i)11-s − 0.188i·13-s + (−0.439 − 0.760i)16-s + (−0.491 + 0.851i)17-s + (1.71 − 0.989i)19-s + 2.23·20-s + 2.55·22-s + (−0.737 + 0.425i)23-s + (−0.151 + 0.262i)25-s + (−0.162 − 0.281i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.415832702\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.415832702\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-4.21 + 2.43i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-6.38 - 11.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-46.8 - 27.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 8.85iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (34.4 - 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-141. + 81.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (81.3 - 46.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 119. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (85.6 + 49.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.01T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-28.6 - 49.6i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (407. + 235. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (112. - 195. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (370. - 213. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (81.9 - 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 79.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (666. + 384. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (267. + 463. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-12.8 - 22.2i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.91200387003845823468996335436, −10.01527658666902161434481997434, −9.260852578131916788487663494470, −7.44139576485410945756951392085, −6.45850250895490700382952847406, −5.77601526922373001190571970902, −4.53808948457535597450610314706, −3.61158290306576451270610221206, −2.57446855419167866906624838429, −1.53879571524846070291179282857,
1.32239510987665189875527550431, 3.16390478184011578402232533444, 4.18645117827096822503497959748, 5.16889124927043765119359694324, 5.86002349831879799071384950620, 6.74207559511762653709398219607, 7.77224469590565545195755437136, 8.911721667887494909229169679207, 9.635105922301415448967274130509, 11.26196499788136285945612000892