| L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.925 + 1.46i)3-s + (0.173 − 0.984i)4-s + (0.569 − 1.56i)5-s + (0.232 + 1.71i)6-s + (−2.36 − 4.10i)7-s + (−0.500 − 0.866i)8-s + (−1.28 − 2.70i)9-s + (−0.569 − 1.56i)10-s + 0.121i·11-s + (1.28 + 1.16i)12-s + (0.927 + 2.54i)13-s + (−4.45 − 1.62i)14-s + (1.76 + 2.28i)15-s + (−0.939 − 0.342i)16-s + (1.15 − 3.17i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (−0.534 + 0.845i)3-s + (0.0868 − 0.492i)4-s + (0.254 − 0.700i)5-s + (0.0947 + 0.700i)6-s + (−0.895 − 1.55i)7-s + (−0.176 − 0.306i)8-s + (−0.428 − 0.903i)9-s + (−0.180 − 0.495i)10-s + 0.0365i·11-s + (0.369 + 0.336i)12-s + (0.257 + 0.707i)13-s + (−1.18 − 0.433i)14-s + (0.455 + 0.589i)15-s + (−0.234 − 0.0855i)16-s + (0.280 − 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.840420 - 0.948249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.840420 - 0.948249i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (0.925 - 1.46i)T \) |
| 19 | \( 1 + (-0.410 + 4.33i)T \) |
| good | 5 | \( 1 + (-0.569 + 1.56i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.36 + 4.10i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.121iT - 11T^{2} \) |
| 13 | \( 1 + (-0.927 - 2.54i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.15 + 3.17i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.910 + 0.160i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.33 + 7.55i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 6.35iT - 31T^{2} \) |
| 37 | \( 1 - 6.86iT - 37T^{2} \) |
| 41 | \( 1 + (-3.20 + 2.69i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.0557 - 0.316i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.24 + 1.45i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.36 - 6.18i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.0715 + 0.405i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.65 + 3.51i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (6.46 - 7.69i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.87 - 3.24i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.587 + 3.33i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.0602 + 0.165i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-14.5 + 8.38i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.94 + 16.7i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-9.05 - 10.7i)T + (-16.8 + 95.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.28344234238096917106117510165, −10.32957650587745882905254735408, −9.738352897525087967111491163916, −8.903999670156725573645876924940, −7.12213202456709418030073497477, −6.26947489244374439329587580375, −4.96021142865747087665967483471, −4.26661742227827436816326184332, −3.20180639173105315587761120952, −0.808339643390833277911548978100,
2.27432716454439851430028334227, 3.36485825452816319684202384172, 5.38755803803406909662518579368, 5.99951481691260353423792698801, 6.61750471050294294785315241185, 7.82493047035811455936785821375, 8.716529489573797579683678148301, 10.04312729706661811162210340143, 11.05805869858096608772645215039, 12.13782872838533172023127137778
