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 | \( 1 + (-1.03 - 7.17i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.53082652633649056496191421426, −11.70729739849939009956289014430, −11.40424028294802956469526646742, −10.17676250110423131348140943589, −9.294685138798800012291738445706, −7.70950542152350954312221663828, −6.77644941151592851053876294394, −6.22011790606740366933409193011, −3.97837959856324366040330949603, −1.89290993201748908837538907324,
1.31641359573412542343833701690, 3.83719798466757548392497230868, 5.29925624134385364483494857606, 6.42474557454744457094205727385, 8.315958114720265616715336359213, 8.932886838120396890075249346473, 9.755902508584352644246340749137, 11.11795287699247332206980127968, 11.98232971058606194957513342309, 12.70550034289339783724606934596