| L(s) = 1 | + (2.43 − 1.40i)2-s + (−2.49 + 4.55i)3-s + (−0.0626 + 0.108i)4-s + (−2.5 − 4.33i)5-s + (0.319 + 14.5i)6-s + (−12.1 + 13.9i)7-s + 22.8i·8-s + (−14.5 − 22.7i)9-s + (−12.1 − 7.01i)10-s + (−24.0 − 13.9i)11-s + (−0.338 − 0.556i)12-s + 85.9i·13-s + (−10.0 + 50.9i)14-s + (25.9 − 0.569i)15-s + (31.4 + 54.5i)16-s + (18.2 − 31.6i)17-s + ⋯ |
| L(s) = 1 | + (0.859 − 0.496i)2-s + (−0.480 + 0.876i)3-s + (−0.00783 + 0.0135i)4-s + (−0.223 − 0.387i)5-s + (0.0217 + 0.991i)6-s + (−0.657 + 0.753i)7-s + 1.00i·8-s + (−0.537 − 0.843i)9-s + (−0.384 − 0.221i)10-s + (−0.660 − 0.381i)11-s + (−0.00813 − 0.0133i)12-s + 1.83i·13-s + (−0.191 + 0.973i)14-s + (0.447 − 0.00980i)15-s + (0.492 + 0.852i)16-s + (0.260 − 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.634145 + 1.04253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.634145 + 1.04253i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.49 - 4.55i)T \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
| 7 | \( 1 + (12.1 - 13.9i)T \) |
| good | 2 | \( 1 + (-2.43 + 1.40i)T + (4 - 6.92i)T^{2} \) |
| 11 | \( 1 + (24.0 + 13.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 85.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-18.2 + 31.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (55.3 - 31.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-100. + 58.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 80.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-268. - 155. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (41.0 + 71.0i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 85.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 94.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-199. - 345. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (121. + 70.2i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (443. - 768. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-98.2 + 56.7i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-130. + 225. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 390. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (183. + 105. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-529. - 916. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (490. + 850. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.69e3iT - 9.12e5T^{2} \) |
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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
−13.51278463680938350139348591273, −12.30924629058009195805309844393, −11.81918805608140695192363554071, −10.74419812716321317030651277774, −9.344656311138076380837881774914, −8.508604642855989362368532475178, −6.38731432592007607465302571131, −5.11955206261205035578099227457, −4.18335302342029017566168857003, −2.81329539794593940251750383040,
0.54274738426863843686560425590, 3.18214307236009092166027942561, 4.94101952830881961367612794597, 6.06867296786011080345334296085, 7.03156683928883792843291283022, 7.980093518508733577303301750448, 10.07870564415266359607356523926, 10.79356809502569831218595873156, 12.42228596831685668122403925184, 13.10241613649376993184003987649
