L(s) = 1 | − 2-s + (−1.5 + 2.59i)3-s + 4-s + (−0.5 + 0.866i)5-s + (1.5 − 2.59i)6-s − 8-s + (−3 − 5.19i)9-s + (0.5 − 0.866i)10-s − 6·11-s + (−1.5 + 2.59i)12-s + (−1.5 − 2.59i)15-s + 16-s + (2 + 3.46i)17-s + (3 + 5.19i)18-s + (2 − 3.46i)19-s + (−0.5 + 0.866i)20-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.866 + 1.49i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.612 − 1.06i)6-s − 0.353·8-s + (−1 − 1.73i)9-s + (0.158 − 0.273i)10-s − 1.80·11-s + (−0.433 + 0.749i)12-s + (−0.387 − 0.670i)15-s + 0.250·16-s + (0.485 + 0.840i)17-s + (0.707 + 1.22i)18-s + (0.458 − 0.794i)19-s + (−0.111 + 0.193i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (6.5 - 0.866i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.5 - 11.2i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.82329198021321592103326677486, −10.11740796518777914487880407003, −9.472298557554296097755407749912, −8.292209352422757863040016787814, −7.34390571778061140995842898880, −5.94134191805903154512109648994, −5.25767330633036476134860173760, −4.03152599555149067529829374220, −2.77937321113669562370556187255, 0,
1.44056956029966402223138470886, 2.83020870303270225348206797819, 5.14456401471591198803257239420, 5.78867838504659961623646506543, 7.03941804960068795595043853125, 7.70392866215363538328473552958, 8.249860879405008774862463864305, 9.647153288400384047556688445816, 10.68806091808231158796079668089