| L(s) = 1 | + (67.7 + 39.0i)3-s + (594. − 342. i)5-s + (−222. − 386. i)9-s + (−1.22e4 + 2.12e4i)11-s + 275. i·13-s + 5.36e4·15-s + (−1.33e5 − 7.72e4i)17-s + (9.69e3 − 5.59e3i)19-s + (−1.42e5 − 2.47e5i)23-s + (3.99e4 − 6.91e4i)25-s − 5.47e5i·27-s − 2.66e5·29-s + (−1.28e6 − 7.39e5i)31-s + (−1.66e6 + 9.58e5i)33-s + (1.24e6 + 2.15e6i)37-s + ⋯ |
| L(s) = 1 | + (0.836 + 0.482i)3-s + (0.950 − 0.548i)5-s + (−0.0339 − 0.0588i)9-s + (−0.837 + 1.45i)11-s + 0.00964i·13-s + 1.05·15-s + (−1.60 − 0.924i)17-s + (0.0743 − 0.0429i)19-s + (−0.510 − 0.883i)23-s + (0.102 − 0.177i)25-s − 1.03i·27-s − 0.376·29-s + (−1.38 − 0.800i)31-s + (−1.40 + 0.808i)33-s + (0.664 + 1.15i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4450192062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4450192062\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-67.7 - 39.0i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-594. + 342. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.22e4 - 2.12e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 275. iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (1.33e5 + 7.72e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.69e3 + 5.59e3i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.42e5 + 2.47e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 2.66e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (1.28e6 + 7.39e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.24e6 - 2.15e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 7.08e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.28e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (1.44e6 - 8.36e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-9.80e5 + 1.69e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.09e7 + 6.34e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-4.25e6 + 2.45e6i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.33e7 - 2.31e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 5.66e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-3.59e7 - 2.07e7i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.42e7 + 2.47e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + 6.35e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (1.00e7 - 5.81e6i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 8.14e6iT - 7.83e15T^{2} \) |
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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
−10.22700123550938249764094537779, −9.506988097894220782573681567592, −8.908873958606744255960827767215, −7.70859334407262988903371535208, −6.46566195244383620949070423867, −5.10525100850251129884758137145, −4.25987330759173664259709956162, −2.64650278933647078254809038299, −1.91009598292149557442077760203, −0.07538476642107885427619910805,
1.74562234020320066155250526580, 2.51443341628692231630249031346, 3.59680346051389892555250676378, 5.43928692448404603789026420996, 6.28490428707835301153637359371, 7.51908427883514838513005331889, 8.474408041298958678639229504465, 9.261177797742757021594081254037, 10.63351799896223810908986935884, 11.09603275566962266645341124874
