| L(s) = 1 | + (−2.10 + 1.88i)2-s + (−1.35 − 5.01i)3-s + (0.892 − 7.95i)4-s + (−5.18 + 14.2i)5-s + (12.3 + 8.01i)6-s + (−9.21 − 1.62i)7-s + (13.1 + 18.4i)8-s + (−23.3 + 13.6i)9-s + (−15.9 − 39.8i)10-s + (59.3 − 21.6i)11-s + (−41.0 + 6.31i)12-s + (44.8 − 37.6i)13-s + (22.4 − 13.9i)14-s + (78.4 + 6.66i)15-s + (−62.4 − 14.1i)16-s + (50.3 − 29.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 + 0.666i)2-s + (−0.261 − 0.965i)3-s + (0.111 − 0.993i)4-s + (−0.463 + 1.27i)5-s + (0.838 + 0.545i)6-s + (−0.497 − 0.0877i)7-s + (0.579 + 0.815i)8-s + (−0.863 + 0.504i)9-s + (−0.503 − 1.25i)10-s + (1.62 − 0.592i)11-s + (−0.988 + 0.152i)12-s + (0.956 − 0.802i)13-s + (0.429 − 0.266i)14-s + (1.35 + 0.114i)15-s + (−0.975 − 0.221i)16-s + (0.718 − 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.937526 + 0.0467141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.937526 + 0.0467141i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.10 - 1.88i)T \) |
| 3 | \( 1 + (1.35 + 5.01i)T \) |
| good | 5 | \( 1 + (5.18 - 14.2i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (9.21 + 1.62i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-59.3 + 21.6i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-44.8 + 37.6i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-50.3 + 29.0i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.1 - 26.6i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-2.42 - 13.7i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (12.2 - 14.5i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-197. + 34.8i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-101. - 175. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-197. - 235. i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (182. + 501. i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-39.9 + 226. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 462. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-56.3 - 20.5i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (45.0 - 255. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (160. + 190. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-533. - 923. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-228. + 396. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-441. + 526. i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (894. + 750. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (114. + 65.9i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.01e3 + 370. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.59759239253132577758121876561, −11.85917660421663875924786329575, −11.16860496694487696671436497477, −10.04010642299361295989005372962, −8.596872695564467936630484954979, −7.52574883755461733609856316395, −6.60854636787625076677542098927, −5.91671709959472735639026027633, −3.24084715061288231948817309048, −0.980649448920153485547451802324,
1.07557658855429611349707418691, 3.65993467903476287386028249563, 4.51489548856444500570784261852, 6.44075292306110532749384106105, 8.265936685343919078425950144397, 9.224581415464895511089110939449, 9.663750824066547511234281247203, 11.20179099998384322383915378345, 11.93052394361109940122759298175, 12.71455436463911073713070184098
