| L(s) = 1 | + (−6.56 − 3.79i)5-s + (−2.04e3 − 1.25e3i)7-s + (1.12e4 + 1.94e4i)11-s − 5.77e3i·13-s + (8.00e4 − 4.61e4i)17-s + (−7.80e4 − 4.50e4i)19-s + (−1.19e4 + 2.06e4i)23-s + (−1.95e5 − 3.38e5i)25-s − 8.51e5·29-s + (1.97e5 − 1.13e5i)31-s + (8.70e3 + 1.59e4i)35-s + (−5.99e5 + 1.03e6i)37-s − 1.51e6i·41-s + 1.73e6·43-s + (7.27e6 + 4.20e6i)47-s + ⋯ |
| L(s) = 1 | + (−0.0105 − 0.00606i)5-s + (−0.853 − 0.521i)7-s + (0.767 + 1.33i)11-s − 0.202i·13-s + (0.958 − 0.553i)17-s + (−0.598 − 0.345i)19-s + (−0.0426 + 0.0738i)23-s + (−0.499 − 0.865i)25-s − 1.20·29-s + (0.213 − 0.123i)31-s + (0.00579 + 0.0106i)35-s + (−0.320 + 0.554i)37-s − 0.536i·41-s + 0.506·43-s + (1.49 + 0.861i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.292380268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292380268\) |
| \(L(5)\) |
|
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 | \( 1 + (2.04e3 + 1.25e3i)T \) |
| good | 5 | \( 1 + (6.56 + 3.79i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.12e4 - 1.94e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 5.77e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-8.00e4 + 4.61e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.80e4 + 4.50e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.19e4 - 2.06e4i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 8.51e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.97e5 + 1.13e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (5.99e5 - 1.03e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.51e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.73e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-7.27e6 - 4.20e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (2.30e6 + 3.98e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (8.79e6 - 5.08e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.60e4 - 9.23e3i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (8.92e5 + 1.54e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.19e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (1.71e7 - 9.89e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-3.53e6 + 6.12e6i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 4.30e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.73e7 - 5.04e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.38e8iT - 7.83e15T^{2} \) |
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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
−10.66125376791012434148463180948, −9.816428581733569605181281325824, −9.160512987660518788646218749877, −7.69840000517708082992104091958, −6.95460621780477746792598137644, −5.93931220961592585802375438619, −4.55318719664059410798211478022, −3.62856486079684337593661575326, −2.29681534978599285246856123335, −0.928919000427370479703371172084,
0.33065221784918580809798818465, 1.68062576954721798142149698252, 3.15717027077593946634882892118, 3.91774588489387329057863628251, 5.67098487178921562122161348517, 6.16329981171654105393361035417, 7.44165513391731919801219535724, 8.648299991024699716793937183504, 9.309969432289248027475815163360, 10.38829222153799943485783012453
