| L(s) = 1 | + (−4.45 + 3.24i)2-s + (6.91 − 21.2i)4-s + (−7.22 + 8.53i)5-s − 10.2·7-s + (24.5 + 75.4i)8-s + (4.58 − 61.4i)10-s + (30.7 − 22.3i)11-s + (68.6 + 49.8i)13-s + (45.7 − 33.2i)14-s + (−208. − 151. i)16-s + (0.731 + 2.24i)17-s + (18.4 + 56.7i)19-s + (131. + 212. i)20-s + (−64.6 + 198. i)22-s + (−121. + 88.4i)23-s + ⋯ |
| L(s) = 1 | + (−1.57 + 1.14i)2-s + (0.864 − 2.66i)4-s + (−0.646 + 0.763i)5-s − 0.554·7-s + (1.08 + 3.33i)8-s + (0.145 − 1.94i)10-s + (0.841 − 0.611i)11-s + (1.46 + 1.06i)13-s + (0.874 − 0.635i)14-s + (−3.26 − 2.37i)16-s + (0.0104 + 0.0321i)17-s + (0.222 + 0.685i)19-s + (1.47 + 2.38i)20-s + (−0.626 + 1.92i)22-s + (−1.10 + 0.802i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0964478 - 0.309067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0964478 - 0.309067i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (7.22 - 8.53i)T \) |
| good | 2 | \( 1 + (4.45 - 3.24i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 + (-30.7 + 22.3i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-68.6 - 49.8i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-0.731 - 2.24i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-18.4 - 56.7i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (121. - 88.4i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (19.2 - 59.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-30.7 - 94.7i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (241. + 175. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (166. + 120. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 223.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-72.4 + 223. i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (166. - 513. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (554. + 402. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-8.59 + 6.24i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-125. - 387. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (43.1 - 132. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (692. - 503. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (7.25 - 22.3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (242. + 746. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (604. - 438. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-132. + 407. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.83853317504896113220866556668, −11.06607330002596690640613924165, −10.22713320459807379673360542636, −9.154915164313788178439675706090, −8.471817710340210739928771360039, −7.39944268969057075776517543593, −6.52120393613172643525782982416, −5.90302492041356016229250821815, −3.73521983858448574886029987695, −1.49153654672776356766793140634,
0.25156576682437814232268392516, 1.43872188455526008900395939337, 3.17554535110048411932096932267, 4.18163534730233999897422918842, 6.48079335810147930507058552940, 7.76316138932392479723821087162, 8.494731055880048122634064133045, 9.288623537167953308434885705935, 10.16522616169717874346077810396, 11.14492205455139517996412250287
