| L(s) = 1 | + (−0.981 − 0.189i)2-s + (0.0497 + 1.04i)3-s + (0.928 + 0.371i)4-s + (0.362 − 0.508i)5-s + (0.148 − 1.03i)6-s + (0.966 + 2.46i)7-s + (−0.841 − 0.540i)8-s + (1.89 − 0.181i)9-s + (−0.452 + 0.431i)10-s + (−4.02 + 0.776i)11-s + (−0.341 + 0.988i)12-s + (2.41 + 0.710i)13-s + (−0.482 − 2.60i)14-s + (0.549 + 0.353i)15-s + (0.723 + 0.690i)16-s + (0.525 + 0.412i)17-s + ⋯ |
| L(s) = 1 | + (−0.694 − 0.133i)2-s + (0.0287 + 0.602i)3-s + (0.464 + 0.185i)4-s + (0.162 − 0.227i)5-s + (0.0607 − 0.422i)6-s + (0.365 + 0.930i)7-s + (−0.297 − 0.191i)8-s + (0.632 − 0.0604i)9-s + (−0.142 + 0.136i)10-s + (−1.21 + 0.234i)11-s + (−0.0987 + 0.285i)12-s + (0.670 + 0.196i)13-s + (−0.128 − 0.695i)14-s + (0.141 + 0.0911i)15-s + (0.180 + 0.172i)16-s + (0.127 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.869763 + 0.537215i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869763 + 0.537215i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.981 + 0.189i)T \) |
| 7 | \( 1 + (-0.966 - 2.46i)T \) |
| 23 | \( 1 + (-0.676 - 4.74i)T \) |
| good | 3 | \( 1 + (-0.0497 - 1.04i)T + (-2.98 + 0.285i)T^{2} \) |
| 5 | \( 1 + (-0.362 + 0.508i)T + (-1.63 - 4.72i)T^{2} \) |
| 11 | \( 1 + (4.02 - 0.776i)T + (10.2 - 4.08i)T^{2} \) |
| 13 | \( 1 + (-2.41 - 0.710i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.525 - 0.412i)T + (4.00 + 16.5i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.810i)T + (4.47 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-0.207 + 1.44i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (5.61 - 2.89i)T + (17.9 - 25.2i)T^{2} \) |
| 37 | \( 1 + (-7.83 + 0.748i)T + (36.3 - 7.00i)T^{2} \) |
| 41 | \( 1 + (-2.65 - 5.81i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.78 + 2.43i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (5.36 + 9.28i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.819 + 3.37i)T + (-47.1 - 24.2i)T^{2} \) |
| 59 | \( 1 + (-1.38 + 1.32i)T + (2.80 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.456 + 9.58i)T + (-60.7 - 5.79i)T^{2} \) |
| 67 | \( 1 + (2.53 + 7.32i)T + (-52.6 + 41.4i)T^{2} \) |
| 71 | \( 1 + (-2.00 + 2.31i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (4.30 + 1.72i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (-0.168 - 0.694i)T + (-70.2 + 36.1i)T^{2} \) |
| 83 | \( 1 + (-6.24 + 13.6i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (15.5 + 8.00i)T + (51.6 + 72.4i)T^{2} \) |
| 97 | \( 1 + (5.20 + 11.3i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.47709928950948454717502923681, −10.80530890787864228582039655944, −9.762544721436405973276752087722, −9.174193276511059789548299201462, −8.175811442067672648929990171930, −7.23565024020536781422024015187, −5.76587951836286452815765894918, −4.85012651989257731594536457310, −3.29856751181675821474737048048, −1.77839522801159404897219574332,
1.00496630679556093513527429560, 2.58062671742389796159765876429, 4.31496253958588909979774408878, 5.80342310730341497396433210546, 6.90495762289835795824921841061, 7.68883042170370222414072869761, 8.339416365746095696414910060896, 9.727153571042460703485288625035, 10.58988428913467844098606084133, 11.06697803166316270647968532643
