| L(s) = 1 | + (−349. − 202. i)5-s + (−1.82e3 − 1.56e3i)7-s + (1.05e4 − 6.10e3i)11-s + 7.95e3·13-s + (−1.40e5 + 8.10e4i)17-s + (−3.50e4 + 6.07e4i)19-s + (−9.31e4 − 5.37e4i)23-s + (−1.13e5 − 1.96e5i)25-s − 8.13e5i·29-s + (−8.76e5 − 1.51e6i)31-s + (3.21e5 + 9.15e5i)35-s + (−1.67e6 + 2.90e6i)37-s − 1.96e6i·41-s + 3.97e6·43-s + (3.18e6 + 1.83e6i)47-s + ⋯ |
| L(s) = 1 | + (−0.559 − 0.323i)5-s + (−0.758 − 0.651i)7-s + (0.722 − 0.417i)11-s + 0.278·13-s + (−1.68 + 0.970i)17-s + (−0.269 + 0.466i)19-s + (−0.332 − 0.192i)23-s + (−0.291 − 0.504i)25-s − 1.15i·29-s + (−0.948 − 1.64i)31-s + (0.214 + 0.609i)35-s + (−0.896 + 1.55i)37-s − 0.695i·41-s + 1.16·43-s + (0.652 + 0.376i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7196267182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7196267182\) |
| \(L(5)\) |
|
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 | \( 1 + (1.82e3 + 1.56e3i)T \) |
| good | 5 | \( 1 + (349. + 202. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.05e4 + 6.10e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 7.95e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + (1.40e5 - 8.10e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (3.50e4 - 6.07e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (9.31e4 + 5.37e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 8.13e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (8.76e5 + 1.51e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.67e6 - 2.90e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 1.96e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 3.97e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-3.18e6 - 1.83e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (4.53e6 - 2.61e6i)T + (3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.47e7 + 8.50e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-7.69e6 + 1.33e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.19e6 + 2.07e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.94e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (-2.58e5 - 4.47e5i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.00e7 - 3.47e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 3.77e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-8.00e7 - 4.62e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.54e8T + 7.83e15T^{2} \) |
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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
−10.82968116694359195611251205357, −9.805329387061705405871565993742, −8.781508688912198644794579402303, −7.944530859282221940040469038179, −6.69269447837240285597253245546, −5.99141281567661232445579387722, −4.18685736182297391318729854530, −3.84986296245149066088816850310, −2.17387018200367175241247105271, −0.70903172753653415074792069838,
0.22145138496470182160820960212, 1.90956680316362761899828284190, 3.10578447486091512218886358786, 4.14041600615861107491642587329, 5.41111505073850371077770683477, 6.71717949148148706378140578789, 7.21781392733707460245220565632, 8.900312399208338285708045113443, 9.160490103445743541838562793174, 10.59710117507591131949973855905
