| L(s) = 1 | + (−1.99 + 0.192i)2-s + (−1.61 + 0.935i)3-s + (3.92 − 0.766i)4-s − 4.90·5-s + (3.04 − 2.17i)6-s + (−10.4 − 6.04i)7-s + (−7.66 + 2.28i)8-s + (−2.75 + 4.76i)9-s + (9.75 − 0.944i)10-s + (−1.28 + 0.741i)11-s + (−5.64 + 4.91i)12-s + (9.39 − 8.98i)13-s + (21.9 + 10.0i)14-s + (7.93 − 4.58i)15-s + (14.8 − 6.02i)16-s + (−11.4 + 19.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 + 0.0963i)2-s + (−0.539 + 0.311i)3-s + (0.981 − 0.191i)4-s − 0.980·5-s + (0.507 − 0.362i)6-s + (−1.49 − 0.862i)7-s + (−0.958 + 0.285i)8-s + (−0.305 + 0.529i)9-s + (0.975 − 0.0944i)10-s + (−0.116 + 0.0674i)11-s + (−0.470 + 0.409i)12-s + (0.722 − 0.690i)13-s + (1.57 + 0.715i)14-s + (0.529 − 0.305i)15-s + (0.926 − 0.376i)16-s + (−0.673 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00109726 - 0.0123823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00109726 - 0.0123823i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.99 - 0.192i)T \) |
| 13 | \( 1 + (-9.39 + 8.98i)T \) |
| good | 3 | \( 1 + (1.61 - 0.935i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + 4.90T + 25T^{2} \) |
| 7 | \( 1 + (10.4 + 6.04i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (1.28 - 0.741i)T + (60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (11.4 - 19.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.00 - 4.61i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (1.28 - 0.743i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (11.9 + 20.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 39.7iT - 961T^{2} \) |
| 37 | \( 1 + (9.45 + 16.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (25.3 + 43.9i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (36.5 + 21.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 3.33iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-8.77 - 5.06i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (22.1 - 38.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (50.9 - 29.3i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (93.8 + 54.1i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + 28.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-35.6 - 61.6i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (40.8 - 70.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−16.03253536742485469386297519752, −15.35107430502618115403415413314, −13.41276550174805210433387816090, −12.06806367873841885577122535883, −10.78102428494770624948237327306, −10.21314893780261238332618333070, −8.566090955178297559043905181422, −7.33418166898639137943795537538, −6.01325915482624321339352988380, −3.59765246552777176920712656226,
0.01700775630883149369413299422, 3.20287515905204634540255675342, 6.11854816522839187271852192026, 7.07371898385355155120572086286, 8.738609543150868703577937381549, 9.618813268047340048446726490872, 11.43681283646746790312424871104, 11.83567007015446469402756437518, 13.07623747074726975434810506453, 15.22006938225735635937438499173
