| L(s) = 1 | + (6.29 + 8.66i)2-s + (3.74 + 11.5i)3-s + (−15.6 + 48.1i)4-s + (−15.2 − 11.0i)5-s + (−76.3 + 105. i)6-s + (51.6 + 16.7i)7-s + (135. − 44.1i)8-s + (470. − 342. i)9-s − 201. i·10-s + (−903. − 977. i)11-s − 614.·12-s + (−502. − 692. i)13-s + (179. + 553. i)14-s + (70.5 − 217. i)15-s + (3.86e3 + 2.80e3i)16-s + (−3.99e3 + 5.50e3i)17-s + ⋯ |
| L(s) = 1 | + (0.786 + 1.08i)2-s + (0.138 + 0.427i)3-s + (−0.244 + 0.752i)4-s + (−0.121 − 0.0885i)5-s + (−0.353 + 0.486i)6-s + (0.150 + 0.0489i)7-s + (0.265 − 0.0861i)8-s + (0.645 − 0.469i)9-s − 0.201i·10-s + (−0.678 − 0.734i)11-s − 0.355·12-s + (−0.228 − 0.315i)13-s + (0.0654 + 0.201i)14-s + (0.0209 − 0.0643i)15-s + (0.942 + 0.684i)16-s + (−0.813 + 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.46904 + 1.29316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46904 + 1.29316i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (903. + 977. i)T \) |
| good | 2 | \( 1 + (-6.29 - 8.66i)T + (-19.7 + 60.8i)T^{2} \) |
| 3 | \( 1 + (-3.74 - 11.5i)T + (-589. + 428. i)T^{2} \) |
| 5 | \( 1 + (15.2 + 11.0i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-51.6 - 16.7i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (502. + 692. i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (3.99e3 - 5.50e3i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (819. - 266. i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 + 2.07e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-2.18e4 - 7.09e3i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (-8.06e3 + 5.86e3i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (3.36e3 - 1.03e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-8.59e4 + 2.79e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 - 2.64e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.15e4 - 1.27e5i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (1.20e4 - 8.74e3i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (9.29e4 - 2.86e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-2.18e5 + 3.01e5i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 + 1.89e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (3.33e5 + 2.42e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-1.02e5 - 3.32e4i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-3.12e5 - 4.30e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (4.35e5 - 6.00e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + 4.93e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.03e6 + 7.53e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.65325989279788762300713398470, −17.83972038843330276702498901494, −16.14862019685893166708372411331, −15.42795160142189238927154421349, −14.12088316622268712798134929366, −12.69132176339141260856130398509, −10.37662155907104217481051208601, −8.104025035091103511647528570364, −6.16414089974262031256065680519, −4.27793958127516703184450647715,
2.19606707367393275855320116956, 4.58040313150546929258061243395, 7.51863661399151453673230300148, 10.15982145629858015080199517135, 11.70308809520237939138555693341, 12.95504239600929672234203920199, 14.01250202712820034766080666141, 15.89690280801652343300720968252, 17.93686939159715215304504177978, 19.30094597359002844920540347297
