| L(s) = 1 | − 252i·3-s + 4.83e3i·5-s + 1.67e4·7-s + 1.13e5·9-s + 5.34e5i·11-s + 5.77e5i·13-s + 1.21e6·15-s − 6.90e6·17-s − 1.06e7i·19-s − 4.21e6i·21-s − 1.86e7·23-s + 2.54e7·25-s − 7.32e7i·27-s − 1.28e8i·29-s − 5.28e7·31-s + ⋯ |
| L(s) = 1 | − 0.598i·3-s + 0.691i·5-s + 0.376·7-s + 0.641·9-s + 1.00i·11-s + 0.431i·13-s + 0.413·15-s − 1.17·17-s − 0.987i·19-s − 0.225i·21-s − 0.603·23-s + 0.522·25-s − 0.982i·27-s − 1.16i·29-s − 0.331·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.197386658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197386658\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 252iT - 1.77e5T^{2} \) |
| 5 | \( 1 - 4.83e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 - 1.67e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 5.34e5iT - 2.85e11T^{2} \) |
| 13 | \( 1 - 5.77e5iT - 1.79e12T^{2} \) |
| 17 | \( 1 + 6.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.06e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 1.86e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.28e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + 5.28e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.82e8iT - 1.77e17T^{2} \) |
| 41 | \( 1 + 3.08e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.71e7iT - 9.29e17T^{2} \) |
| 47 | \( 1 - 2.68e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.59e9iT - 9.26e18T^{2} \) |
| 59 | \( 1 + 5.18e9iT - 3.01e19T^{2} \) |
| 61 | \( 1 + 6.95e9iT - 4.35e19T^{2} \) |
| 67 | \( 1 - 1.54e10iT - 1.22e20T^{2} \) |
| 71 | \( 1 + 9.79e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.46e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.81e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.93e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 2.49e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 7.50e10T + 7.15e21T^{2} \) |
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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
−10.00377301217573836702793710041, −9.035918656374600845697955894721, −7.79033683178480988145693039390, −6.99565904177631937222669027263, −6.42331077006785696948417421072, −4.85337946903035835533733557232, −4.02216309695980623188817407932, −2.42238912355134045656027474980, −1.85703657132688955076378683499, −0.49747825090964987213239593180,
0.795404227159861550504585360639, 1.77606892777127041953409347500, 3.27673451371320770854529727683, 4.27481368932643670188059330801, 5.09617586295505686955744056456, 6.12288393029425129222884992473, 7.41975118486474722585633121860, 8.521439484605351261879212200072, 9.110651470307590233323603198794, 10.32394876768805268643001915499
