| L(s) = 1 | + (−428. − 247. i)5-s + (1.93e3 + 1.42e3i)7-s + (1.06e3 + 1.83e3i)11-s − 1.56e4i·13-s + (4.89e4 − 2.82e4i)17-s + (−1.40e5 − 8.08e4i)19-s + (−2.23e5 + 3.87e5i)23-s + (−7.27e4 − 1.26e5i)25-s + 1.33e6·29-s + (−4.67e5 + 2.69e5i)31-s + (−4.76e5 − 1.08e6i)35-s + (5.31e5 − 9.21e5i)37-s + 4.94e6i·41-s + 2.05e6·43-s + (2.47e6 + 1.42e6i)47-s + ⋯ |
| L(s) = 1 | + (−0.685 − 0.396i)5-s + (0.805 + 0.592i)7-s + (0.0725 + 0.125i)11-s − 0.547i·13-s + (0.586 − 0.338i)17-s + (−1.07 − 0.620i)19-s + (−0.799 + 1.38i)23-s + (−0.186 − 0.322i)25-s + 1.88·29-s + (−0.506 + 0.292i)31-s + (−0.317 − 0.725i)35-s + (0.283 − 0.491i)37-s + 1.75i·41-s + 0.601·43-s + (0.506 + 0.292i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.244170114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244170114\) |
| \(L(5)\) |
|
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 | \( 1 + (-1.93e3 - 1.42e3i)T \) |
| good | 5 | \( 1 + (428. + 247. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.06e3 - 1.83e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.56e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-4.89e4 + 2.82e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.40e5 + 8.08e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.23e5 - 3.87e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.33e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (4.67e5 - 2.69e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-5.31e5 + 9.21e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 - 4.94e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.05e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.47e6 - 1.42e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.97e6 + 1.20e7i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-5.87e6 + 3.39e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-4.80e5 - 2.77e5i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.71e7 + 2.97e7i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 2.21e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (1.19e7 - 6.89e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.94e7 + 3.37e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 6.23e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-4.64e7 - 2.68e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 2.92e7iT - 7.83e15T^{2} \) |
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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
−10.40686854860453923516953789862, −9.256112485650840403846118841592, −8.240551314656287352545369434606, −7.70707267770646093310150258514, −6.26734394598400168622491268270, −5.11845360932819929177735335376, −4.26293833836719711570702387345, −2.89444530079654634901613968646, −1.57725410790620373047722597464, −0.30194462883120642584371418615,
1.03894431110472344517144617378, 2.33821630179525288280431195310, 3.84413117929332960180427787645, 4.48705439605453732165569013836, 5.98274403619480552373574546417, 7.05971375848194159311346012546, 7.984760271584304931710896905792, 8.719503026259715212843435843336, 10.26267370441256117584330275974, 10.77194246436721519184407786293
