| L(s) = 1 | + (−0.284 − 1.97i)2-s + (−1.97 + 4.33i)3-s + (−3.83 + 1.12i)4-s + (−7.35 + 8.48i)5-s + (9.13 + 2.68i)6-s + (−14.4 + 9.27i)7-s + (3.32 + 7.27i)8-s + (2.82 + 3.26i)9-s + (18.8 + 12.1i)10-s + (0.679 − 4.72i)11-s + (2.71 − 18.8i)12-s + (12.9 + 8.31i)13-s + (22.4 + 25.9i)14-s + (−22.2 − 48.6i)15-s + (13.4 − 8.65i)16-s + (−25.6 − 7.53i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.380 + 0.833i)3-s + (−0.479 + 0.140i)4-s + (−0.657 + 0.758i)5-s + (0.621 + 0.182i)6-s + (−0.779 + 0.500i)7-s + (0.146 + 0.321i)8-s + (0.104 + 0.120i)9-s + (0.597 + 0.383i)10-s + (0.0186 − 0.129i)11-s + (0.0652 − 0.453i)12-s + (0.276 + 0.177i)13-s + (0.429 + 0.495i)14-s + (−0.382 − 0.837i)15-s + (0.210 − 0.135i)16-s + (−0.365 − 0.107i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.431243 + 0.532354i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.431243 + 0.532354i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.284 + 1.97i)T \) |
| 23 | \( 1 + (-4.38 + 110. i)T \) |
| good | 3 | \( 1 + (1.97 - 4.33i)T + (-17.6 - 20.4i)T^{2} \) |
| 5 | \( 1 + (7.35 - 8.48i)T + (-17.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (14.4 - 9.27i)T + (142. - 312. i)T^{2} \) |
| 11 | \( 1 + (-0.679 + 4.72i)T + (-1.27e3 - 374. i)T^{2} \) |
| 13 | \( 1 + (-12.9 - 8.31i)T + (912. + 1.99e3i)T^{2} \) |
| 17 | \( 1 + (25.6 + 7.53i)T + (4.13e3 + 2.65e3i)T^{2} \) |
| 19 | \( 1 + (45.5 - 13.3i)T + (5.77e3 - 3.70e3i)T^{2} \) |
| 29 | \( 1 + (-285. - 83.7i)T + (2.05e4 + 1.31e4i)T^{2} \) |
| 31 | \( 1 + (-88.7 - 194. i)T + (-1.95e4 + 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-161. - 185. i)T + (-7.20e3 + 5.01e4i)T^{2} \) |
| 41 | \( 1 + (-55.7 + 64.3i)T + (-9.80e3 - 6.82e4i)T^{2} \) |
| 43 | \( 1 + (133. - 291. i)T + (-5.20e4 - 6.00e4i)T^{2} \) |
| 47 | \( 1 + 440.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-504. + 324. i)T + (6.18e4 - 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-411. - 264. i)T + (8.53e4 + 1.86e5i)T^{2} \) |
| 61 | \( 1 + (281. + 615. i)T + (-1.48e5 + 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-76.5 - 532. i)T + (-2.88e5 + 8.47e4i)T^{2} \) |
| 71 | \( 1 + (99.7 + 693. i)T + (-3.43e5 + 1.00e5i)T^{2} \) |
| 73 | \( 1 + (546. - 160. i)T + (3.27e5 - 2.10e5i)T^{2} \) |
| 79 | \( 1 + (596. + 383. i)T + (2.04e5 + 4.48e5i)T^{2} \) |
| 83 | \( 1 + (-54.9 - 63.3i)T + (-8.13e4 + 5.65e5i)T^{2} \) |
| 89 | \( 1 + (231. - 507. i)T + (-4.61e5 - 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-36.9 + 42.6i)T + (-1.29e5 - 9.03e5i)T^{2} \) |
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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
−15.73209734578337060549622429566, −14.60042051913691031806529070555, −13.06800967765038769264816170257, −11.82111798624031976043916891034, −10.78424865879344501567618850680, −9.963659177910124125246876827103, −8.511636709588712296624122763259, −6.57640428230284285213849971220, −4.59643960174693870523253233309, −3.08460548295097084547560322847,
0.59490690753019530181128110066, 4.21940248051429744315262016032, 6.13592731786561678845314627923, 7.23216264888220428792824973213, 8.449575771655650194288649916285, 9.924729831565221821727008620159, 11.73388825609335310101893472636, 12.82334682791966917645301487457, 13.53470859248690075078201351510, 15.27029627825205179575228391298
