| L(s) = 1 | + (−4 + 6.92i)2-s + (−13.7 − 23.8i)3-s + (−31.9 − 55.4i)4-s + (−121. − 210. i)5-s + 220.·6-s + (208. + 360. i)7-s + 511.·8-s + (715. − 1.23e3i)9-s + 1.94e3·10-s + 4.33e3·11-s + (−880. + 1.52e3i)12-s + (−3.13e3 − 5.43e3i)13-s − 3.32e3·14-s + (−3.33e3 + 5.78e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−2.31e3 + 4.01e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.294 − 0.509i)3-s + (−0.249 − 0.433i)4-s + (−0.434 − 0.752i)5-s + 0.415·6-s + (0.229 + 0.397i)7-s + 0.353·8-s + (0.327 − 0.566i)9-s + 0.614·10-s + 0.983·11-s + (−0.147 + 0.254i)12-s + (−0.395 − 0.685i)13-s − 0.324·14-s + (−0.255 + 0.442i)15-s + (−0.125 + 0.216i)16-s + (−0.114 + 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0918907 - 0.469389i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0918907 - 0.469389i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 37 | \( 1 + (2.99e5 - 7.31e4i)T \) |
| good | 3 | \( 1 + (13.7 + 23.8i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (121. + 210. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-208. - 360. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 4.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.13e3 + 5.43e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (2.31e3 - 4.01e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-5.77e3 - 1.00e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + 1.08e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.25e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (2.83e5 + 4.91e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + 4.19e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.90e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + (4.24e5 - 7.35e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.49e6 - 2.58e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.49e5 - 2.58e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (4.21e5 + 7.29e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-9.93e5 - 1.72e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 1.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-3.92e6 - 6.79e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-7.76e5 + 1.34e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.28e6 + 2.22e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 6.18e6T + 8.07e13T^{2} \) |
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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
−12.33399219982534323733263523611, −11.98310505485932239545782716263, −10.18450862679698061454208039797, −8.951729656748321263020902900231, −7.990738530653003916142114102146, −6.73363331917225936855429062474, −5.56700556866558605427578971815, −4.05524484062806988367497785786, −1.50389217266244190050390878927, −0.19870318377841487995455244270,
1.80349279069331571907097187564, 3.60202122850075881896939314543, 4.69075583361971147633901665449, 6.72393908685735298408586584090, 7.87608447420464397679000527137, 9.410673124927170393699457326219, 10.38794496829533104637433400315, 11.28673059866810894537941045978, 12.06995292009200113678006369471, 13.67572867705205614826767996495
