| L(s) = 1 | + (−5.04 + 2.55i)2-s + 11.5·3-s + (18.9 − 25.7i)4-s + (−58.5 + 29.5i)6-s + 231. i·7-s + (−30.0 + 178. i)8-s − 108.·9-s + 559. i·11-s + (219. − 298. i)12-s + 107.·13-s + (−590. − 1.16e3i)14-s + (−303. − 977. i)16-s + 441. i·17-s + (548. − 277. i)18-s − 1.87e3i·19-s + ⋯ |
| L(s) = 1 | + (−0.892 + 0.451i)2-s + 0.743·3-s + (0.592 − 0.805i)4-s + (−0.663 + 0.335i)6-s + 1.78i·7-s + (−0.165 + 0.986i)8-s − 0.446·9-s + 1.39i·11-s + (0.440 − 0.598i)12-s + 0.177·13-s + (−0.805 − 1.59i)14-s + (−0.296 − 0.954i)16-s + 0.370i·17-s + (0.398 − 0.201i)18-s − 1.19i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6716416100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6716416100\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.04 - 2.55i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 11.5T + 243T^{2} \) |
| 7 | \( 1 - 231. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 559. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 107.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 441. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.87e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.83e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.36e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 7.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 925.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.06e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 7.95e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.68e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.12e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.86e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.55e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.24e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.86e4iT - 8.58e9T^{2} \) |
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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
−12.01825351054369374678812896575, −11.01498695182620066800736725101, −9.709505743513171607877363726068, −8.924587665840541515396400012921, −8.458262857864866604489214746768, −7.21261622778084979891005911433, −6.05886391486352594682718731717, −4.95692067465304246690006057623, −2.71839675305500625471228959059, −1.97784609373732301002911592379,
0.23960735741956002725790517334, 1.47135038708260513291377563273, 3.22044400870789855389239442479, 3.83148637623010211377805912362, 6.03153114549514602243089593124, 7.49299097817008402312845822834, 7.981805517683938546827164414106, 9.080844584322692670454182948273, 10.00443087305689585254661265657, 10.97916862338663793933676464076
