| L(s) = 1 | + (31.6 − 45.2i)2-s + (6.96 − 79.5i)3-s + (−691. − 1.89e3i)4-s + (−137. + 3.12e3i)5-s + (−3.37e3 − 2.83e3i)6-s + (1.12e4 − 3.01e3i)7-s + (−5.31e4 − 1.42e4i)8-s + (5.18e4 + 9.14e3i)9-s + (1.36e5 + 1.05e5i)10-s + (−7.14e4 − 1.23e5i)11-s + (−1.55e5 + 4.17e4i)12-s + (3.11e4 − 2.72e3i)13-s + (2.19e5 − 6.03e5i)14-s + (2.47e5 + 3.27e4i)15-s + (−7.40e5 + 6.21e5i)16-s + (9.45e5 − 1.35e6i)17-s + ⋯ |
| L(s) = 1 | + (0.989 − 1.41i)2-s + (0.0286 − 0.327i)3-s + (−0.675 − 1.85i)4-s + (−0.0441 + 0.999i)5-s + (−0.434 − 0.364i)6-s + (0.668 − 0.179i)7-s + (−1.62 − 0.434i)8-s + (0.878 + 0.154i)9-s + (1.36 + 1.05i)10-s + (−0.443 − 0.768i)11-s + (−0.626 + 0.167i)12-s + (0.0840 − 0.00735i)13-s + (0.408 − 1.12i)14-s + (0.325 + 0.0430i)15-s + (−0.706 + 0.592i)16-s + (0.665 − 0.951i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0586i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.998 + 0.0586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.110091 - 3.75244i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.110091 - 3.75244i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (137. - 3.12e3i)T \) |
| 19 | \( 1 + (1.80e6 - 1.69e6i)T \) |
| good | 2 | \( 1 + (-31.6 + 45.2i)T + (-350. - 962. i)T^{2} \) |
| 3 | \( 1 + (-6.96 + 79.5i)T + (-5.81e4 - 1.02e4i)T^{2} \) |
| 7 | \( 1 + (-1.12e4 + 3.01e3i)T + (2.44e8 - 1.41e8i)T^{2} \) |
| 11 | \( 1 + (7.14e4 + 1.23e5i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-3.11e4 + 2.72e3i)T + (1.35e11 - 2.39e10i)T^{2} \) |
| 17 | \( 1 + (-9.45e5 + 1.35e6i)T + (-6.89e11 - 1.89e12i)T^{2} \) |
| 23 | \( 1 + (-3.92e6 + 8.41e6i)T + (-2.66e13 - 3.17e13i)T^{2} \) |
| 29 | \( 1 + (-7.69e6 - 1.35e6i)T + (3.95e14 + 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-7.29e6 + 1.26e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + (5.42e7 + 5.42e7i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + (3.88e6 - 3.25e6i)T + (2.33e15 - 1.32e16i)T^{2} \) |
| 43 | \( 1 + (2.26e8 - 1.05e8i)T + (1.38e16 - 1.65e16i)T^{2} \) |
| 47 | \( 1 + (-1.34e8 + 9.43e7i)T + (1.79e16 - 4.94e16i)T^{2} \) |
| 53 | \( 1 + (-4.58e7 + 9.84e7i)T + (-1.12e17 - 1.33e17i)T^{2} \) |
| 59 | \( 1 + (-6.67e8 + 1.17e8i)T + (4.80e17 - 1.74e17i)T^{2} \) |
| 61 | \( 1 + (-1.24e9 + 4.54e8i)T + (5.46e17 - 4.58e17i)T^{2} \) |
| 67 | \( 1 + (3.59e8 - 2.51e8i)T + (6.23e17 - 1.71e18i)T^{2} \) |
| 71 | \( 1 + (1.75e8 + 6.37e7i)T + (2.49e18 + 2.09e18i)T^{2} \) |
| 73 | \( 1 + (1.18e7 - 1.35e8i)T + (-4.23e18 - 7.46e17i)T^{2} \) |
| 79 | \( 1 + (-2.07e8 - 2.47e8i)T + (-1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-1.44e9 - 5.38e9i)T + (-1.34e19 + 7.75e18i)T^{2} \) |
| 89 | \( 1 + (-5.11e9 + 6.09e9i)T + (-5.41e18 - 3.07e19i)T^{2} \) |
| 97 | \( 1 + (5.01e9 - 7.16e9i)T + (-2.52e19 - 6.92e19i)T^{2} \) |
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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
−11.45995589418179289181901656683, −10.66146445402192022448183983003, −9.979957988829873443078461844320, −8.079389152663362328405152608428, −6.71358343025413179794512640324, −5.27106815260249772307695702290, −4.07999432823185902604120631250, −2.93376165713874974619035408102, −1.93340426741414769935819987958, −0.68104716695830224240012607808,
1.51408866961074417998140390654, 3.76576002562712946838270104054, 4.75697276210189626108805930241, 5.34456037762523264608917608356, 6.82036616401349445110911434994, 7.87479664701574771516864877520, 8.775760282833113620875072767477, 10.17980868626930073917769308006, 11.96294166621320732824243188102, 12.86503770417778312735817978659
