| L(s) = 1 | + (0.841 + 0.540i)2-s + (−0.369 − 2.56i)3-s + (0.415 + 0.909i)4-s + (0.959 + 0.281i)5-s + (1.07 − 2.36i)6-s + (1.04 − 1.20i)7-s + (−0.142 + 0.989i)8-s + (−3.58 + 1.05i)9-s + (0.654 + 0.755i)10-s + (1.45 − 0.937i)11-s + (2.18 − 1.40i)12-s + (−2.84 − 3.28i)13-s + (1.53 − 0.450i)14-s + (0.369 − 2.56i)15-s + (−0.654 + 0.755i)16-s + (−0.474 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (−0.213 − 1.48i)3-s + (0.207 + 0.454i)4-s + (0.429 + 0.125i)5-s + (0.440 − 0.964i)6-s + (0.395 − 0.456i)7-s + (−0.0503 + 0.349i)8-s + (−1.19 + 0.351i)9-s + (0.207 + 0.238i)10-s + (0.439 − 0.282i)11-s + (0.630 − 0.405i)12-s + (−0.789 − 0.911i)13-s + (0.409 − 0.120i)14-s + (0.0953 − 0.663i)15-s + (−0.163 + 0.188i)16-s + (−0.115 + 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54615 - 0.671760i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54615 - 0.671760i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (-4.79 - 0.0163i)T \) |
| good | 3 | \( 1 + (0.369 + 2.56i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (-1.04 + 1.20i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.45 + 0.937i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (2.84 + 3.28i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.474 - 1.03i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.47 - 3.22i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.12 - 6.83i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.936 - 6.51i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (5.21 - 1.53i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (2.55 + 0.750i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.942 - 6.55i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + (1.27 - 1.46i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (7.22 + 8.34i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.704 + 4.89i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (3.88 + 2.49i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.30 - 4.05i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-1.34 - 2.95i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (10.2 + 11.8i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (3.60 - 1.05i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (2.05 + 14.2i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (4.93 + 1.44i)T + (81.6 + 52.4i)T^{2} \) |
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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
−12.56010689352938572913277123218, −11.40899292873371503082845186170, −10.40663492720944525269570440966, −8.803362005662187942820858231109, −7.64542955636420357961516711952, −7.06502474015236314885137530441, −6.04303761078744151722834933652, −5.03829108646208108767034319098, −3.15732116355578762531902387984, −1.50275045934147508897087794203,
2.37294634570503414923402917448, 3.99105569929068678788231749896, 4.84092412800208405769739399209, 5.65589405975065982097270358383, 7.10500595402662107125717409209, 9.039906303057953985770269527889, 9.467847778032384070298057072777, 10.45654414531506479203073802712, 11.42205532984095839047709341629, 12.01040222704333953085197474714
