| L(s) = 1 | + (3.46 − 2i)2-s + (9.95 + 11.9i)3-s + (7.99 − 13.8i)4-s + (41.5 + 72.0i)5-s + (58.4 + 21.6i)6-s + (−128. − 15.2i)7-s − 63.9i·8-s + (−44.9 + 238. i)9-s + (288. + 166. i)10-s + (455. + 263. i)11-s + (245. − 41.9i)12-s − 776. i·13-s + (−476. + 204. i)14-s + (−450. + 1.21e3i)15-s + (−128 − 221. i)16-s + (236. − 409. i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.638 + 0.769i)3-s + (0.249 − 0.433i)4-s + (0.744 + 1.28i)5-s + (0.663 + 0.245i)6-s + (−0.993 − 0.117i)7-s − 0.353i·8-s + (−0.184 + 0.982i)9-s + (0.911 + 0.526i)10-s + (1.13 + 0.655i)11-s + (0.492 − 0.0839i)12-s − 1.27i·13-s + (−0.649 + 0.279i)14-s + (−0.516 + 1.39i)15-s + (−0.125 − 0.216i)16-s + (0.198 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.63582 + 0.964384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.63582 + 0.964384i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.46 + 2i)T \) |
| 3 | \( 1 + (-9.95 - 11.9i)T \) |
| 7 | \( 1 + (128. + 15.2i)T \) |
| good | 5 | \( 1 + (-41.5 - 72.0i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-455. - 263. i)T + (8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 776. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + (-236. + 409. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.71e3 + 990. i)T + (1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.49e3 - 863. i)T + (3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 6.04e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + (7.50e3 + 4.33e3i)T + (1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-1.50e3 - 2.61e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.63e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.19e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.93e3 - 3.35e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.24e3 - 1.87e3i)T + (2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.24e3 + 5.61e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-3.85e4 + 2.22e4i)T + (4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.21e4 - 2.11e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 8.35e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-1.11e4 - 6.41e3i)T + (1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.01e4 + 3.49e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.28e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.92e4 + 5.07e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.01e3iT - 8.58e9T^{2} \) |
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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
−14.92222019511902153311385666716, −14.06551645243701500054580938720, −13.12682863665083909708274394662, −11.39397248150130743083170646151, −10.04088135940735687188022390389, −9.616589036066281380033118348442, −7.21530191020795344088149084624, −5.76221557846472752893220133722, −3.69473379189226860246875555721, −2.60152868956206793374342082424,
1.51120121513796403814305100814, 3.67600093243620725543663626927, 5.74951920659998844506178953976, 6.87553710736478587349155230822, 8.717680605751621267961412066836, 9.402779986374174262024610751867, 11.98552130732788742357072364665, 12.70226427291572458849210643155, 13.75239060250356885112777821269, 14.40141018863389372915508890586
