| L(s) = 1 | + (0.220 + 1.53i)2-s + (−1.10 + 1.27i)3-s + (−0.374 + 0.110i)4-s + (−0.601 + 0.386i)5-s + (−2.18 − 1.40i)6-s + (−0.391 − 2.72i)7-s + (1.03 + 2.26i)8-s + (0.0240 + 0.167i)9-s + (−0.724 − 0.835i)10-s + (3.38 − 2.17i)11-s + (0.273 − 0.598i)12-s + (0.379 − 0.832i)13-s + (4.07 − 1.19i)14-s + (0.171 − 1.19i)15-s + (−3.89 + 2.50i)16-s + (−0.709 − 0.208i)17-s + ⋯ |
| L(s) = 1 | + (0.155 + 1.08i)2-s + (−0.636 + 0.734i)3-s + (−0.187 + 0.0550i)4-s + (−0.269 + 0.172i)5-s + (−0.893 − 0.574i)6-s + (−0.147 − 1.02i)7-s + (0.365 + 0.800i)8-s + (0.00802 + 0.0558i)9-s + (−0.229 − 0.264i)10-s + (1.02 − 0.656i)11-s + (0.0788 − 0.172i)12-s + (0.105 − 0.230i)13-s + (1.09 − 0.320i)14-s + (0.0442 − 0.307i)15-s + (−0.973 + 0.625i)16-s + (−0.172 − 0.0505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534893 + 0.683904i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534893 + 0.683904i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (-6.96 + 4.29i)T \) |
| good | 2 | \( 1 + (-0.220 - 1.53i)T + (-1.91 + 0.563i)T^{2} \) |
| 3 | \( 1 + (1.10 - 1.27i)T + (-0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.601 - 0.386i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.391 + 2.72i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-3.38 + 2.17i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.379 + 0.832i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (0.709 + 0.208i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.962 + 6.69i)T + (-18.2 - 5.35i)T^{2} \) |
| 23 | \( 1 + (1.79 - 2.07i)T + (-3.27 - 22.7i)T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + (1.93 + 4.23i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + (-5.27 - 1.54i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-8.65 - 2.53i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (7.37 - 8.51i)T + (-6.68 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-7.29 + 2.14i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.81 + 10.5i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.10 - 0.710i)T + (25.3 + 55.4i)T^{2} \) |
| 71 | \( 1 + (-0.936 + 0.274i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.98 - 1.91i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (4.62 - 10.1i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (12.0 - 7.73i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-4.71 - 5.43i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−15.47759475821212769175960348915, −14.25690217287278392876370771172, −13.39005342604714011637273860254, −11.31823451701822054648560942243, −10.95916542606449091686955826176, −9.391519644494838705350936787660, −7.73856535464920871275338181922, −6.69839712123786170945392016473, −5.41294864253551256224147896572, −4.04353129471180137993356740970,
1.81691943013905374783199452849, 3.89270894046844294842534682873, 5.96497745018869722875971425292, 7.15198949665793631243349567539, 8.958773332924471143403776556001, 10.23749667180604605446209351265, 11.73220563417770065166779036287, 12.11731169153202036644179693878, 12.71480533346396638495460265573, 14.29754260035803746846789097121
