| L(s) = 1 | + (0.591 − 1.82i)2-s + (−7.58 − 5.51i)3-s + (3.50 + 2.54i)4-s + (−4.63 − 10.1i)5-s + (−14.5 + 10.5i)6-s + 14.5·7-s + (19.1 − 13.8i)8-s + (18.8 + 57.9i)9-s + (−21.2 + 2.42i)10-s + (7.88 − 24.2i)11-s + (−12.5 − 38.6i)12-s + (5.22 + 16.0i)13-s + (8.59 − 26.4i)14-s + (−20.8 + 102. i)15-s + (−3.24 − 9.99i)16-s + (−58.3 + 42.3i)17-s + ⋯ |
| L(s) = 1 | + (0.209 − 0.643i)2-s + (−1.45 − 1.06i)3-s + (0.438 + 0.318i)4-s + (−0.414 − 0.909i)5-s + (−0.987 + 0.717i)6-s + 0.784·7-s + (0.844 − 0.613i)8-s + (0.697 + 2.14i)9-s + (−0.672 + 0.0766i)10-s + (0.216 − 0.664i)11-s + (−0.302 − 0.930i)12-s + (0.111 + 0.342i)13-s + (0.164 − 0.505i)14-s + (−0.359 + 1.76i)15-s + (−0.0507 − 0.156i)16-s + (−0.832 + 0.604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.503701 - 0.815421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503701 - 0.815421i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.63 + 10.1i)T \) |
| good | 2 | \( 1 + (-0.591 + 1.82i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (7.58 + 5.51i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 14.5T + 343T^{2} \) |
| 11 | \( 1 + (-7.88 + 24.2i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-5.22 - 16.0i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (58.3 - 42.3i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-62.4 + 45.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-16.3 + 50.3i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-173. - 125. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (56.0 - 40.6i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-93.5 - 287. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-0.229 - 0.707i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 93.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-86.0 - 62.5i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (334. + 243. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (12.5 + 38.4i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-138. + 425. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (344. - 250. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (32.5 + 23.6i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-147. + 455. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-99.4 - 72.2i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (904. - 657. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (482. - 1.48e3i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.14e3 - 831. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.84201541060095579479206741611, −15.99028813225812663154585671601, −13.49340642462213842379048572463, −12.49767075404162975085243892050, −11.61052781636935671789303684341, −10.96181144354130104773541846036, −8.156482860190653678925729037371, −6.67700683332612105710344268925, −4.82420723878832659656927625749, −1.31106848400525078922777251392,
4.57877162224634835682482955687, 5.95301292194301146419080688843, 7.30307614484638011118434811425, 10.04148118371654493418429707176, 11.08701248893977165175991899563, 11.76183430139419181267124261334, 14.37062709936856026398052062826, 15.38193610458017882120062132645, 15.98277695624403370443312657743, 17.31965741815782860149955351926
