| L(s) = 1 | + (0.203 + 0.235i)2-s + (−13.1 − 8.42i)3-s + (9.09 − 63.2i)4-s + (171. + 78.3i)5-s + (−0.689 − 4.79i)6-s + (−84.9 + 289. i)7-s + (33.4 − 21.5i)8-s + (100. + 221. i)9-s + (16.5 + 56.2i)10-s + (205. + 177. i)11-s + (−652. + 752. i)12-s + (209. − 61.6i)13-s + (−85.3 + 38.9i)14-s + (−1.58e3 − 2.47e3i)15-s + (−3.91e3 − 1.14e3i)16-s + (7.05e3 − 1.01e3i)17-s + ⋯ |
| L(s) = 1 | + (0.0254 + 0.0293i)2-s + (−0.485 − 0.312i)3-s + (0.142 − 0.988i)4-s + (1.37 + 0.626i)5-s + (−0.00319 − 0.0222i)6-s + (−0.247 + 0.843i)7-s + (0.0653 − 0.0420i)8-s + (0.138 + 0.303i)9-s + (0.0165 + 0.0562i)10-s + (0.154 + 0.133i)11-s + (−0.377 + 0.435i)12-s + (0.0955 − 0.0280i)13-s + (−0.0310 + 0.0142i)14-s + (−0.471 − 0.733i)15-s + (−0.955 − 0.280i)16-s + (1.43 − 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.12744 - 0.258560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12744 - 0.258560i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.1 + 8.42i)T \) |
| 23 | \( 1 + (-1.20e4 - 1.50e3i)T \) |
| good | 2 | \( 1 + (-0.203 - 0.235i)T + (-9.10 + 63.3i)T^{2} \) |
| 5 | \( 1 + (-171. - 78.3i)T + (1.02e4 + 1.18e4i)T^{2} \) |
| 7 | \( 1 + (84.9 - 289. i)T + (-9.89e4 - 6.36e4i)T^{2} \) |
| 11 | \( 1 + (-205. - 177. i)T + (2.52e5 + 1.75e6i)T^{2} \) |
| 13 | \( 1 + (-209. + 61.6i)T + (4.06e6 - 2.60e6i)T^{2} \) |
| 17 | \( 1 + (-7.05e3 + 1.01e3i)T + (2.31e7 - 6.80e6i)T^{2} \) |
| 19 | \( 1 + (-5.91e3 - 849. i)T + (4.51e7 + 1.32e7i)T^{2} \) |
| 29 | \( 1 + (1.95e3 + 1.35e4i)T + (-5.70e8 + 1.67e8i)T^{2} \) |
| 31 | \( 1 + (7.15e3 - 4.60e3i)T + (3.68e8 - 8.07e8i)T^{2} \) |
| 37 | \( 1 + (-3.98e4 + 1.81e4i)T + (1.68e9 - 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-1.82e4 + 3.98e4i)T + (-3.11e9 - 3.58e9i)T^{2} \) |
| 43 | \( 1 + (-7.08e4 + 1.10e5i)T + (-2.62e9 - 5.75e9i)T^{2} \) |
| 47 | \( 1 + 1.02e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (3.17e4 - 1.08e5i)T + (-1.86e10 - 1.19e10i)T^{2} \) |
| 59 | \( 1 + (1.39e5 - 4.10e4i)T + (3.54e10 - 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-3.75e4 - 5.83e4i)T + (-2.14e10 + 4.68e10i)T^{2} \) |
| 67 | \( 1 + (1.80e5 - 1.56e5i)T + (1.28e10 - 8.95e10i)T^{2} \) |
| 71 | \( 1 + (-7.84e4 - 9.05e4i)T + (-1.82e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (7.51e3 - 5.22e4i)T + (-1.45e11 - 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-1.96e5 - 6.67e5i)T + (-2.04e11 + 1.31e11i)T^{2} \) |
| 83 | \( 1 + (7.07e5 - 3.23e5i)T + (2.14e11 - 2.47e11i)T^{2} \) |
| 89 | \( 1 + (4.20e5 - 6.54e5i)T + (-2.06e11 - 4.52e11i)T^{2} \) |
| 97 | \( 1 + (-1.22e6 - 5.57e5i)T + (5.45e11 + 6.29e11i)T^{2} \) |
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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
−13.68122305933575217808653944525, −12.36081608362816602011793651683, −11.09972178708475339890697395823, −10.02166022984970178366199973145, −9.307348939121099949194121303279, −7.12962196828178189187500684738, −5.89911600299059086557433313562, −5.44817978642253715421345441286, −2.58596141495565907070429726139, −1.23089344774373556235002655621,
1.17329644786624573848144802198, 3.25518184727629209190491956032, 4.86334141180265125970679797271, 6.21410893735836345753572918680, 7.60558461402995778961491176269, 9.155828380157239950695313921675, 10.06665017696242305149909277562, 11.35326778881629256032067969989, 12.65301127172356404809204240099, 13.31301218684442340669759657418
