| L(s) = 1 | + (−7.75 − 13.5i)3-s + 6.84i·5-s − 49i·7-s + (−122. + 209. i)9-s − 604.·11-s + 655.·13-s + (92.5 − 53.1i)15-s + 59.9i·17-s + 1.24e3i·19-s + (−662. + 380. i)21-s − 1.66e3·23-s + 3.07e3·25-s + (3.78e3 + 32.0i)27-s + 1.12e3i·29-s − 1.58e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.497 − 0.867i)3-s + 0.122i·5-s − 0.377i·7-s + (−0.504 + 0.863i)9-s − 1.50·11-s + 1.07·13-s + (0.106 − 0.0609i)15-s + 0.0502i·17-s + 0.792i·19-s + (−0.327 + 0.188i)21-s − 0.656·23-s + 0.984·25-s + (0.999 + 0.00846i)27-s + 0.248i·29-s − 0.295i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.343353198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343353198\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (7.75 + 13.5i)T \) |
| 7 | \( 1 + 49iT \) |
| good | 5 | \( 1 - 6.84iT - 3.12e3T^{2} \) |
| 11 | \( 1 + 604.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 655.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 59.9iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.24e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 1.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.12e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.58e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 260.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.48e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 3.84e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.33e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.24e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.29e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.14e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 5.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.10e3iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 2.61e4T + 8.58e9T^{2} \) |
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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
−10.78485233506221806918195220083, −9.983036572893449541587908717819, −8.362487738708985395663186290169, −7.86844363499497362117139340260, −6.74572568790104828265871433306, −5.86857077924912542839565019730, −4.86450971408447688262769703594, −3.28665863358178081811712108075, −1.95558881184947694929657576637, −0.67564927307184622166613934696,
0.61234602970863494732195101327, 2.53417853804570517064519878585, 3.74297677740616867730511364895, 4.98672021114473108266603300056, 5.64857104435603217764596078803, 6.79342835060156675695444017145, 8.223929352613262037323245108324, 8.945551552059488995451608595862, 10.05099501203521027948672581018, 10.76472313730308440790029139151
