| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.654 + 0.755i)3-s + (0.841 + 0.540i)4-s + (0.455 − 3.17i)5-s + (0.841 − 0.540i)6-s + (0.628 + 1.37i)7-s + (−0.654 − 0.755i)8-s + (−0.142 − 0.989i)9-s + (−1.33 + 2.91i)10-s + (6.27 − 1.84i)11-s + (−0.959 + 0.281i)12-s + (1.19 − 2.62i)13-s + (−0.215 − 1.49i)14-s + (2.09 + 2.42i)15-s + (0.415 + 0.909i)16-s + (−1.00 + 0.646i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.378 + 0.436i)3-s + (0.420 + 0.270i)4-s + (0.203 − 1.41i)5-s + (0.343 − 0.220i)6-s + (0.237 + 0.520i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (−0.420 + 0.921i)10-s + (1.89 − 0.555i)11-s + (−0.276 + 0.0813i)12-s + (0.332 − 0.728i)13-s + (−0.0575 − 0.400i)14-s + (0.541 + 0.625i)15-s + (0.103 + 0.227i)16-s + (−0.244 + 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.765438 - 0.252454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765438 - 0.252454i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 + 0.281i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.66 + 1.10i)T \) |
| good | 5 | \( 1 + (-0.455 + 3.17i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.628 - 1.37i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-6.27 + 1.84i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.19 + 2.62i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (1.00 - 0.646i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.467 - 0.300i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (7.29 - 4.68i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.41 - 5.09i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.206 - 1.43i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.262 + 1.82i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (1.42 - 1.64i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + (-5.01 - 10.9i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-1.64 + 3.60i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (5.91 + 6.82i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-7.35 - 2.16i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-9.07 - 2.66i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (0.527 + 0.339i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (5.72 - 12.5i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.21 - 15.4i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-3.67 + 4.24i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.03 + 7.17i)T + (-93.0 - 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.70550034289339783724606934596, −11.98232971058606194957513342309, −11.11795287699247332206980127968, −9.755902508584352644246340749137, −8.932886838120396890075249346473, −8.315958114720265616715336359213, −6.42474557454744457094205727385, −5.29925624134385364483494857606, −3.83719798466757548392497230868, −1.31641359573412542343833701690,
1.89290993201748908837538907324, 3.97837959856324366040330949603, 6.22011790606740366933409193011, 6.77644941151592851053876294394, 7.70950542152350954312221663828, 9.294685138798800012291738445706, 10.17676250110423131348140943589, 11.40424028294802956469526646742, 11.70729739849939009956289014430, 13.53082652633649056496191421426
