| L(s) = 1 | + (0.998 − 0.641i)2-s + (3.84 − 26.7i)3-s + (−52.5 + 115. i)4-s + (115. − 33.9i)5-s + (−13.3 − 29.1i)6-s + (93.0 + 107. i)7-s + (42.9 + 298. i)8-s + (−699. − 205. i)9-s + (93.6 − 108. i)10-s + (−1.16e3 − 748. i)11-s + (2.87e3 + 1.84e3i)12-s + (−880. + 1.01e3i)13-s + (161. + 47.4i)14-s + (−463. − 3.22e3i)15-s + (−1.03e4 − 1.19e4i)16-s + (−5.38e3 − 1.17e4i)17-s + ⋯ |
| L(s) = 1 | + (0.0882 − 0.0566i)2-s + (0.0821 − 0.571i)3-s + (−0.410 + 0.899i)4-s + (0.413 − 0.121i)5-s + (−0.0251 − 0.0550i)6-s + (0.102 + 0.118i)7-s + (0.0296 + 0.206i)8-s + (−0.319 − 0.0939i)9-s + (0.0296 − 0.0341i)10-s + (−0.263 − 0.169i)11-s + (0.480 + 0.308i)12-s + (−0.111 + 0.128i)13-s + (0.0157 + 0.00462i)14-s + (−0.0354 − 0.246i)15-s + (−0.633 − 0.730i)16-s + (−0.265 − 0.581i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.162288 + 0.528387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162288 + 0.528387i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (3.88e4 + 4.34e4i)T \) |
| good | 2 | \( 1 + (-0.998 + 0.641i)T + (53.1 - 116. i)T^{2} \) |
| 5 | \( 1 + (-115. + 33.9i)T + (6.57e4 - 4.22e4i)T^{2} \) |
| 7 | \( 1 + (-93.0 - 107. i)T + (-1.17e5 + 8.15e5i)T^{2} \) |
| 11 | \( 1 + (1.16e3 + 748. i)T + (8.09e6 + 1.77e7i)T^{2} \) |
| 13 | \( 1 + (880. - 1.01e3i)T + (-8.93e6 - 6.21e7i)T^{2} \) |
| 17 | \( 1 + (5.38e3 + 1.17e4i)T + (-2.68e8 + 3.10e8i)T^{2} \) |
| 19 | \( 1 + (2.16e4 - 4.73e4i)T + (-5.85e8 - 6.75e8i)T^{2} \) |
| 29 | \( 1 + (-5.62e4 - 1.23e5i)T + (-1.12e10 + 1.30e10i)T^{2} \) |
| 31 | \( 1 + (-3.62e4 - 2.52e5i)T + (-2.63e10 + 7.75e9i)T^{2} \) |
| 37 | \( 1 + (5.53e5 + 1.62e5i)T + (7.98e10 + 5.13e10i)T^{2} \) |
| 41 | \( 1 + (3.50e5 - 1.02e5i)T + (1.63e11 - 1.05e11i)T^{2} \) |
| 43 | \( 1 + (2.12e4 - 1.47e5i)T + (-2.60e11 - 7.65e10i)T^{2} \) |
| 47 | \( 1 + 1.16e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.10e6 - 1.27e6i)T + (-1.67e11 + 1.16e12i)T^{2} \) |
| 59 | \( 1 + (-2.43e5 + 2.81e5i)T + (-3.54e11 - 2.46e12i)T^{2} \) |
| 61 | \( 1 + (1.25e5 + 8.73e5i)T + (-3.01e12 + 8.85e11i)T^{2} \) |
| 67 | \( 1 + (-8.80e5 + 5.65e5i)T + (2.51e12 - 5.51e12i)T^{2} \) |
| 71 | \( 1 + (-2.85e6 + 1.83e6i)T + (3.77e12 - 8.27e12i)T^{2} \) |
| 73 | \( 1 + (-1.78e5 + 3.89e5i)T + (-7.23e12 - 8.34e12i)T^{2} \) |
| 79 | \( 1 + (-2.87e6 + 3.32e6i)T + (-2.73e12 - 1.90e13i)T^{2} \) |
| 83 | \( 1 + (6.55e5 + 1.92e5i)T + (2.28e13 + 1.46e13i)T^{2} \) |
| 89 | \( 1 + (-3.27e5 + 2.27e6i)T + (-4.24e13 - 1.24e13i)T^{2} \) |
| 97 | \( 1 + (6.88e6 - 2.02e6i)T + (6.79e13 - 4.36e13i)T^{2} \) |
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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
−13.66031224291867406000661029564, −12.58566654605810541076277735280, −11.92832180293139828779212616992, −10.34665302851796976858241779980, −8.861834105365756781858400018628, −8.026814032643663189531632962649, −6.68876051482561435876837142200, −5.11200615911454385233375862801, −3.46451880417203111962068199684, −1.91595880319029082387088948904,
0.17550406727478930112507950174, 2.11923329108812861303991601754, 4.16145947477576085685400742273, 5.33181714669272717169970303873, 6.53755198473087619355284221059, 8.387837171381352992581455477716, 9.640189129361736598359047944567, 10.33444353239799189935388130594, 11.47234080769515882658503236262, 13.23227302342687085653932802352
