| L(s) = 1 | + (−724. − 418. i)5-s + (613. − 2.32e3i)7-s + (−1.82e4 + 1.05e4i)11-s − 4.89e4·13-s + (5.72e4 − 3.30e4i)17-s + (−1.24e5 + 2.16e5i)19-s + (3.18e4 + 1.84e4i)23-s + (1.54e5 + 2.68e5i)25-s − 1.59e5i·29-s + (4.58e5 + 7.93e5i)31-s + (−1.41e6 + 1.42e6i)35-s + (−6.70e5 + 1.16e6i)37-s − 1.76e6i·41-s − 2.70e6·43-s + (−7.31e6 − 4.22e6i)47-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.669i)5-s + (0.255 − 0.966i)7-s + (−1.24 + 0.720i)11-s − 1.71·13-s + (0.685 − 0.395i)17-s + (−0.957 + 1.65i)19-s + (0.113 + 0.0657i)23-s + (0.396 + 0.686i)25-s − 0.226i·29-s + (0.496 + 0.859i)31-s + (−0.943 + 0.950i)35-s + (−0.357 + 0.619i)37-s − 0.623i·41-s − 0.790·43-s + (−1.50 − 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.5987035895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5987035895\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-613. + 2.32e3i)T \) |
| good | 5 | \( 1 + (724. + 418. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.82e4 - 1.05e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + 4.89e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-5.72e4 + 3.30e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.24e5 - 2.16e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-3.18e4 - 1.84e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 1.59e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-4.58e5 - 7.93e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (6.70e5 - 1.16e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.76e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 2.70e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (7.31e6 + 4.22e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-4.99e6 + 2.88e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-4.25e6 + 2.45e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-4.36e6 + 7.56e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.27e6 + 2.20e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 8.62e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.18e6 - 3.79e6i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.66e7 + 2.89e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.92e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (9.08e7 + 5.24e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.61e8T + 7.83e15T^{2} \) |
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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
−10.34807101274172465646669312437, −9.970714468781465191142666707357, −8.275269789694665157815901347126, −7.78218600561269298835898137707, −6.98878246377324145043059833352, −5.13457466579713316579146112238, −4.56227477392614770601916497909, −3.41910039414098927759984045232, −1.89611155398036268169956750330, −0.41364625277277893821580414347,
0.30032003704790043230212934318, 2.41572542832678487509168591808, 3.02364395328407190757806648837, 4.54651975316899019060841930587, 5.46469126576840131556904362181, 6.83380623075233205991702050068, 7.78635310095566805431533641714, 8.450992563467848886969407624061, 9.725894948601732939129026661481, 10.83770380325627227130151684225
