L(s) = 1 | + (−1.36 + 0.381i)2-s + (−1.60 + 0.430i)3-s + (1.70 − 1.03i)4-s + (1.94 + 0.522i)5-s + (2.02 − 1.20i)6-s + (−1.89 + 1.85i)7-s + (−1.93 + 2.06i)8-s + (−0.196 + 0.113i)9-s + (−2.85 + 0.0319i)10-s + (1.63 + 6.09i)11-s + (−2.30 + 2.40i)12-s + (1.13 + 1.13i)13-s + (1.86 − 3.24i)14-s − 3.35·15-s + (1.84 − 3.55i)16-s + (−0.960 + 1.66i)17-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.269i)2-s + (−0.928 + 0.248i)3-s + (0.854 − 0.519i)4-s + (0.871 + 0.233i)5-s + (0.827 − 0.490i)6-s + (−0.714 + 0.699i)7-s + (−0.682 + 0.730i)8-s + (−0.0655 + 0.0378i)9-s + (−0.902 + 0.0101i)10-s + (0.492 + 1.83i)11-s + (−0.664 + 0.694i)12-s + (0.314 + 0.314i)13-s + (0.499 − 0.866i)14-s − 0.867·15-s + (0.460 − 0.887i)16-s + (−0.233 + 0.403i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.349377 + 0.386824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.349377 + 0.386824i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 - 0.381i)T \) |
| 7 | \( 1 + (1.89 - 1.85i)T \) |
good | 3 | \( 1 + (1.60 - 0.430i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-1.94 - 0.522i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 6.09i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.13 - 1.13i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.960 - 1.66i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 6.09i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.924 - 0.533i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.08 - 5.08i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.198 - 0.343i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.327 - 0.0877i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 7.26iT - 41T^{2} \) |
| 43 | \( 1 + (-1.75 + 1.75i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.08 + 1.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.111 - 0.415i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.32 + 8.67i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (2.72 - 10.1i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-7.87 + 2.10i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 2.78iT - 71T^{2} \) |
| 73 | \( 1 + (-2.05 - 1.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.17 - 1.17i)T + 83iT^{2} \) |
| 89 | \( 1 + (-11.0 + 6.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2.21T + 97T^{2} \) |
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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
−14.08309865827864255817784522800, −12.53343305712420806434283180969, −11.65171549831466274733658751452, −10.49834751828140535456224845259, −9.705270664756684309969187990450, −8.870253984949308659516763823176, −6.98548654620016571630387847264, −6.26266551010151639255438064134, −5.09671595318780222199091983001, −2.25500096370910710768764165627,
0.901714340244421763534196521492, 3.33831491049437256000268987635, 5.96451651277535007446635468329, 6.34498697103080927555736325087, 8.025174549114710229466500294200, 9.240577640005565461125479145423, 10.23019135126167079146829031812, 11.15090969168084391105740374634, 12.01951612460559255970138440808, 13.20218574720820875570515910412