L(s) = 1 | + (−3.89 + 2.24i)2-s + (2.59 + 1.5i)3-s + (6.10 − 10.5i)4-s + (−0.533 − 11.1i)5-s − 13.4·6-s + (−3.71 + 18.1i)7-s + 18.9i·8-s + (4.5 + 7.79i)9-s + (27.1 + 42.2i)10-s + (−11.2 + 19.4i)11-s + (31.7 − 18.3i)12-s + 70.4i·13-s + (−26.3 − 78.9i)14-s + (15.3 − 29.8i)15-s + (6.30 + 10.9i)16-s + (−52.3 − 30.2i)17-s + ⋯ |
L(s) = 1 | + (−1.37 + 0.794i)2-s + (0.499 + 0.288i)3-s + (0.763 − 1.32i)4-s + (−0.0476 − 0.998i)5-s − 0.917·6-s + (−0.200 + 0.979i)7-s + 0.836i·8-s + (0.166 + 0.288i)9-s + (0.859 + 1.33i)10-s + (−0.307 + 0.532i)11-s + (0.763 − 0.440i)12-s + 1.50i·13-s + (−0.502 − 1.50i)14-s + (0.264 − 0.513i)15-s + (0.0985 + 0.170i)16-s + (−0.747 − 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0558864 + 0.511043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0558864 + 0.511043i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.59 - 1.5i)T \) |
| 5 | \( 1 + (0.533 + 11.1i)T \) |
| 7 | \( 1 + (3.71 - 18.1i)T \) |
good | 2 | \( 1 + (3.89 - 2.24i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (11.2 - 19.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 70.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (52.3 + 30.2i)T + (2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (18.2 + 31.5i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (169. - 98.0i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (88.3 - 153. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-190. + 109. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 99.6iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (372. - 214. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-360. - 208. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (385. - 668. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (46.2 + 80.0i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-23.7 - 13.6i)T + (1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 388.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-200. - 115. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (58.7 + 101. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.07e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (93.5 + 162. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 415. iT - 9.12e5T^{2} \) |
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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
−14.01065431169163276082631054456, −12.68910207566785739842695873429, −11.52642704599274516795200788934, −9.883104393131937217746688085605, −9.170517948212627588696619787887, −8.607907081271661601703672593058, −7.45444715244409578204581979347, −6.10145862200540706951969385573, −4.52302636452947791063533608604, −1.91840810636909424136291513780,
0.40469801481149778547341323551, 2.36506933473743784271393035602, 3.58784575376348900311042336511, 6.37697599636279316547642366173, 7.76554593826282808416329641901, 8.264762048736592360220956683717, 9.907494966470214934233067465707, 10.43521698562146810088550891466, 11.24953880956212045923974344868, 12.64247064064999808396980813444