| L(s) = 1 | + (−4.72 + 3.03i)2-s + (−1.28 + 8.90i)3-s + (−0.168 + 0.367i)4-s + (61.1 − 17.9i)5-s + (−21.0 − 46.0i)6-s + (−57.2 − 66.1i)7-s + (−25.9 − 180. i)8-s + (−77.7 − 22.8i)9-s + (−234. + 270. i)10-s + (−265. − 170. i)11-s + (−3.06 − 1.96i)12-s + (422. − 487. i)13-s + (471. + 138. i)14-s + (81.6 + 567. i)15-s + (661. + 763. i)16-s + (56.9 + 124. i)17-s + ⋯ |
| L(s) = 1 | + (−0.835 + 0.537i)2-s + (−0.0821 + 0.571i)3-s + (−0.00525 + 0.0114i)4-s + (1.09 − 0.321i)5-s + (−0.238 − 0.521i)6-s + (−0.441 − 0.509i)7-s + (−0.143 − 0.995i)8-s + (−0.319 − 0.0939i)9-s + (−0.742 + 0.856i)10-s + (−0.661 − 0.424i)11-s + (−0.00613 − 0.00394i)12-s + (0.692 − 0.799i)13-s + (0.643 + 0.188i)14-s + (0.0936 + 0.651i)15-s + (0.646 + 0.746i)16-s + (0.0477 + 0.104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.995366 - 0.0950597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.995366 - 0.0950597i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 8.90i)T \) |
| 23 | \( 1 + (-2.30e3 - 1.07e3i)T \) |
| good | 2 | \( 1 + (4.72 - 3.03i)T + (13.2 - 29.1i)T^{2} \) |
| 5 | \( 1 + (-61.1 + 17.9i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (57.2 + 66.1i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (265. + 170. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (-422. + 487. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-56.9 - 124. i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (-938. + 2.05e3i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (-1.95e3 - 4.28e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (1.15e3 + 8.06e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-8.92e3 - 2.62e3i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (5.33e3 - 1.56e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-684. + 4.75e3i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 + 2.90e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (3.62e3 + 4.18e3i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-1.75e4 + 2.03e4i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (-1.22e3 - 8.51e3i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-3.36e4 + 2.16e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-3.62e4 + 2.32e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (-1.91e4 + 4.18e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (-2.25e4 + 2.59e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (4.81e3 + 1.41e3i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (1.41e3 - 9.86e3i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (1.12e5 - 3.31e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−13.40673989410016595978220320427, −13.10405682218845390749271940756, −11.05217496378931345223385278417, −9.926512059115981949568388614828, −9.226042255402771210270326809241, −8.041396817790672349336959059616, −6.57176133626957938840974251587, −5.24287534526548961308171653431, −3.28369166224738811795542683335, −0.65238846697517032364556653040,
1.41652083423820854801747243429, 2.59929493656974412235059533386, 5.43203143893684914993981171316, 6.51308381910278138242364802039, 8.227481011401997458470362150163, 9.429967041926488530754315816533, 10.18534291309833626495106984373, 11.35549066146952008740424163087, 12.59722257631343569911020103556, 13.75672526592094270299346532380
