| L(s) = 1 | + (3.23 + 2.35i)2-s − 20.9·3-s + (4.94 + 15.2i)4-s + (−11.1 − 34.3i)5-s + (−67.7 − 49.2i)6-s + (105. − 76.4i)7-s + (−19.7 + 60.8i)8-s + 195.·9-s + (44.6 − 137. i)10-s + (−76.2 + 234. i)11-s + (−103. − 318. i)12-s + (617. + 448. i)13-s + 520.·14-s + (233. + 718. i)15-s + (−207. + 150. i)16-s + (−467. + 1.43e3i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s − 1.34·3-s + (0.154 + 0.475i)4-s + (−0.199 − 0.614i)5-s + (−0.768 − 0.558i)6-s + (0.811 − 0.589i)7-s + (−0.109 + 0.336i)8-s + 0.803·9-s + (0.141 − 0.434i)10-s + (−0.189 + 0.584i)11-s + (−0.207 − 0.638i)12-s + (1.01 + 0.736i)13-s + 0.709·14-s + (0.268 + 0.824i)15-s + (−0.202 + 0.146i)16-s + (−0.391 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.37642 + 0.769823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37642 + 0.769823i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 41 | \( 1 + (-6.71e3 - 8.41e3i)T \) |
| good | 3 | \( 1 + 20.9T + 243T^{2} \) |
| 5 | \( 1 + (11.1 + 34.3i)T + (-2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-105. + 76.4i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (76.2 - 234. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-617. - 448. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (467. - 1.43e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-430. + 312. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-3.67e3 - 2.66e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (63.5 + 195. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-145. + 446. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (3.23e3 + 9.94e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 43 | \( 1 + (-6.47e3 - 4.70e3i)T + (4.54e7 + 1.39e8i)T^{2} \) |
| 47 | \( 1 + (-5.81e3 - 4.22e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (3.15e3 + 9.72e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.16e4 - 1.57e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.62e4 - 1.18e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (3.30e3 + 1.01e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.23e4 + 3.79e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + 7.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.89e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.75e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (6.24e4 - 4.53e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.69e4 - 1.44e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.34790214958760562519112215908, −12.47296081505721996278037107718, −11.36315477610605724372357562502, −10.77697850292376753231251034964, −8.862675679380143660515435195185, −7.46796703101227231113348603574, −6.25431653135834758970524433803, −5.06621472743614090762084421687, −4.17879215632794762251027764463, −1.22482731136351865527798451834,
0.810483251312669797410061116711, 2.96576089067473064489638238178, 4.88313326524166118664865607435, 5.73472374043174192067153725432, 6.93939834224041876421298503544, 8.691509652492937302873793834855, 10.58719540697532952223739994827, 11.12760598399233718105294509831, 11.81643787983194587279591138746, 12.96493597523726571394286449733
