| L(s) = 1 | + (0.666 − 0.0957i)2-s + (0.369 + 1.69i)3-s + (−1.48 + 0.435i)4-s + (1.76 − 2.03i)5-s + (0.408 + 1.09i)6-s + (0.360 + 0.560i)7-s + (−2.17 + 0.991i)8-s + (−2.72 + 1.25i)9-s + (0.978 − 1.52i)10-s + (0.634 − 4.41i)11-s + (−1.28 − 2.35i)12-s + (−2.96 − 1.90i)13-s + (0.293 + 0.338i)14-s + (4.09 + 2.22i)15-s + (1.25 − 0.803i)16-s + (−3.35 − 0.984i)17-s + ⋯ |
| L(s) = 1 | + (0.471 − 0.0677i)2-s + (0.213 + 0.976i)3-s + (−0.742 + 0.217i)4-s + (0.787 − 0.909i)5-s + (0.166 + 0.445i)6-s + (0.136 + 0.211i)7-s + (−0.767 + 0.350i)8-s + (−0.908 + 0.416i)9-s + (0.309 − 0.481i)10-s + (0.191 − 1.33i)11-s + (−0.371 − 0.678i)12-s + (−0.822 − 0.528i)13-s + (0.0785 + 0.0905i)14-s + (1.05 + 0.575i)15-s + (0.312 − 0.200i)16-s + (−0.812 − 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05538 + 0.245970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05538 + 0.245970i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.369 - 1.69i)T \) |
| 23 | \( 1 + (-3.88 - 2.81i)T \) |
| good | 2 | \( 1 + (-0.666 + 0.0957i)T + (1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.76 + 2.03i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (-0.360 - 0.560i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.634 + 4.41i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.96 + 1.90i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (3.35 + 0.984i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 5.55i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.554 - 1.88i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.95 - 6.47i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.669 - 0.579i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.34 + 1.16i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.37 + 1.99i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.638iT - 47T^{2} \) |
| 53 | \( 1 + (7.53 - 4.84i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.89 + 6.05i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-10.2 + 4.67i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-6.72 + 0.967i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (11.9 - 1.71i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (11.6 - 3.43i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (2.26 - 3.53i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-2.09 - 2.41i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.93 + 10.7i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.90 - 5.11i)T + (13.8 + 96.0i)T^{2} \) |
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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
−14.56967103058046589788036856771, −13.81428213802410197620894424017, −12.89448295076120245083775180277, −11.62637916827396898055936045486, −10.09757828110376082425713335223, −9.063877063188652962136313801164, −8.393791008925883882786323509348, −5.63251899969292117565835777971, −4.94082060252745485022218070326, −3.32316727625234286863503892320,
2.45737557186924988047694250554, 4.68866790708783819583187993204, 6.36896784881418282667498775513, 7.22129704705860203176143744378, 9.022054259975533427413166336993, 10.00769371672501493380442824480, 11.59979987112716310321837195877, 12.88834482358905191853195853509, 13.56066036519103490673264000888, 14.55684256146216122060978210066
