| L(s) = 1 | + (2.14 − 1.37i)2-s + (−3.84 + 26.7i)3-s + (−50.4 + 110. i)4-s + (−464. + 136. i)5-s + (28.5 + 62.6i)6-s + (1.02e3 + 1.18e3i)7-s + (90.5 + 629. i)8-s + (−699. − 205. i)9-s + (−808. + 932. i)10-s + (3.01e3 + 1.93e3i)11-s + (−2.75e3 − 1.77e3i)12-s + (−3.52e3 + 4.06e3i)13-s + (3.84e3 + 1.12e3i)14-s + (−1.86e3 − 1.29e4i)15-s + (−9.12e3 − 1.05e4i)16-s + (−1.44e4 − 3.17e4i)17-s + ⋯ |
| L(s) = 1 | + (0.189 − 0.121i)2-s + (−0.0821 + 0.571i)3-s + (−0.394 + 0.863i)4-s + (−1.66 + 0.487i)5-s + (0.0540 + 0.118i)6-s + (1.13 + 1.30i)7-s + (0.0625 + 0.434i)8-s + (−0.319 − 0.0939i)9-s + (−0.255 + 0.294i)10-s + (0.682 + 0.438i)11-s + (−0.461 − 0.296i)12-s + (−0.444 + 0.513i)13-s + (0.374 + 0.110i)14-s + (−0.142 − 0.989i)15-s + (−0.556 − 0.642i)16-s + (−0.715 − 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.306670 - 0.769170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.306670 - 0.769170i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.84 - 26.7i)T \) |
| 23 | \( 1 + (-5.08e4 + 2.86e4i)T \) |
| good | 2 | \( 1 + (-2.14 + 1.37i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (464. - 136. i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (-1.02e3 - 1.18e3i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (-3.01e3 - 1.93e3i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (3.52e3 - 4.06e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (1.44e4 + 3.17e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (2.07e4 - 4.54e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-6.86e4 - 1.50e5i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (1.83e4 + 1.27e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (-1.46e5 - 4.31e4i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (1.11e5 - 3.26e4i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (-4.72e4 + 3.28e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 - 6.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (4.89e5 + 5.64e5i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (6.44e5 - 7.43e5i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.27e5 - 1.58e6i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-1.22e5 + 7.88e4i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (-2.25e6 + 1.44e6i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (2.18e6 - 4.78e6i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-1.36e6 + 1.57e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (2.15e6 + 6.32e5i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-1.06e5 + 7.42e5i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (4.62e6 - 1.35e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−14.37343292698228639471367980925, −12.29862662499814542930128457160, −11.84692729470568172286433045337, −11.10745924026086491973549317199, −9.078535710280113648909720506008, −8.290605056477942462289699842219, −7.15255682341389554973713026604, −4.88079852436699135475216408950, −4.08199779389502704777928075192, −2.63774844628752035736676079639,
0.32797914992413983658495853193, 1.19538260011433834940992754620, 4.04811414898251827947216066721, 4.79169899020854313302402760381, 6.69932274643591311792294663765, 7.83973438651176932041060392552, 8.770875554571510826317288821006, 10.80185176153181815864728455806, 11.27860873767890056249796495966, 12.72236770083247258226130144180
