| L(s) = 1 | + (−1.47 + 2.55i)2-s + (11.3 + 10.7i)3-s + (11.6 + 20.1i)4-s + (−32.4 − 56.1i)5-s + (−44.1 + 13.1i)6-s + (80.0 − 138. i)7-s − 163.·8-s + (13.4 + 242. i)9-s + 191.·10-s + (103. − 178. i)11-s + (−84.1 + 352. i)12-s + (32.8 + 56.9i)13-s + (236. + 409. i)14-s + (234. − 983. i)15-s + (−131. + 227. i)16-s − 601.·17-s + ⋯ |
| L(s) = 1 | + (−0.261 + 0.452i)2-s + (0.726 + 0.687i)3-s + (0.363 + 0.629i)4-s + (−0.580 − 1.00i)5-s + (−0.500 + 0.149i)6-s + (0.617 − 1.07i)7-s − 0.901·8-s + (0.0554 + 0.998i)9-s + 0.605·10-s + (0.257 − 0.445i)11-s + (−0.168 + 0.707i)12-s + (0.0539 + 0.0934i)13-s + (0.322 + 0.558i)14-s + (0.269 − 1.12i)15-s + (−0.128 + 0.222i)16-s − 0.504·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.08697 + 0.544038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08697 + 0.544038i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-11.3 - 10.7i)T \) |
| good | 2 | \( 1 + (1.47 - 2.55i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (32.4 + 56.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-80.0 + 138. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-103. + 178. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-32.8 - 56.9i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 601.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.18e3 - 3.78e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-467. + 809. i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.62e3 + 4.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 5.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.22e3 - 5.58e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.40e3 + 2.43e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (3.90e3 - 6.76e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (7.84e3 + 1.35e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.11e4 - 1.93e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-9.45e3 - 1.63e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 4.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.56e4 + 2.70e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.09e4 + 5.35e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (3.67e4 - 6.36e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−20.61274356106632315040728961945, −19.62572271803213014026321578759, −17.18413820591550148178791819490, −16.37410345933802684560187776907, −15.07563006344057724331023149847, −13.24542653827459682195254667136, −11.21911810755312676650161482092, −8.874342402580441188622860750618, −7.71056405824513429845656300894, −4.13734693840406136273485179559,
2.40332837540074594633476136565, 6.70848782916142223454640995462, 8.777365834679306994223103789856, 10.89733283190714772643598390960, 12.29970902841917084296562366338, 14.70869059799347901399816426522, 15.11330140471780305633477749717, 18.14181633768460173281083568618, 18.85480037486389477546965199426, 19.87106248101471328136461937697
