L(s) = 1 | + (4.09 − 3.19i)3-s + (−3.53 + 6.12i)5-s + (7.59 + 16.8i)7-s + (6.56 − 26.1i)9-s + (7.40 − 4.27i)11-s + (45.3 + 26.1i)13-s + (5.08 + 36.3i)15-s − 38.9·17-s + 66.4i·19-s + (85.1 + 44.9i)21-s + (173. + 100. i)23-s + (37.5 + 64.9i)25-s + (−56.7 − 128. i)27-s + (52.9 − 30.5i)29-s + (−116. − 67.2i)31-s + ⋯ |
L(s) = 1 | + (0.788 − 0.615i)3-s + (−0.316 + 0.547i)5-s + (0.410 + 0.912i)7-s + (0.243 − 0.969i)9-s + (0.202 − 0.117i)11-s + (0.967 + 0.558i)13-s + (0.0875 + 0.626i)15-s − 0.555·17-s + 0.801i·19-s + (0.884 + 0.466i)21-s + (1.57 + 0.910i)23-s + (0.300 + 0.519i)25-s + (−0.404 − 0.914i)27-s + (0.339 − 0.195i)29-s + (−0.674 − 0.389i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.422433971\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.422433971\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.09 + 3.19i)T \) |
| 7 | \( 1 + (-7.59 - 16.8i)T \) |
good | 5 | \( 1 + (3.53 - 6.12i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.40 + 4.27i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-45.3 - 26.1i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 38.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 66.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-173. - 100. i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-52.9 + 30.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (116. + 67.2i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-221. + 383. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-26.1 - 45.2i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (137. + 238. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 136. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (191. - 331. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (261. - 151. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-318. + 552. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 228. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.24e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (100. + 174. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (323. + 560. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 826.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-17.0 + 9.85i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.60698476351785811891450832056, −10.97702783596082877891483481795, −9.346554160517510571373506421010, −8.777112569312527517553706355202, −7.74509523854565740721961554665, −6.79432245477421857238836381428, −5.71774622600248876323614468274, −3.96930393527748122020887878805, −2.81732798725772340949102685523, −1.48812880456432851080236141030,
1.03152435422959753923672781415, 2.92603113223569878780778281765, 4.23154065540545566961505178989, 4.89877728578085851424934580268, 6.68743063219829567105602656972, 7.88184399419129696189564185713, 8.621982681214666187114380593818, 9.476588048627646355789500240760, 10.76597448331309193728238377956, 11.12596679956129071683060833776