L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.822 + 1.42i)5-s + (−1.32 + 2.29i)7-s + 0.999·8-s + (−0.822 − 1.42i)10-s + (0.5 + 0.866i)11-s + 5·13-s + (−1.32 − 2.29i)14-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + (2.82 − 4.88i)19-s + 1.64·20-s − 0.999·22-s + (0.822 − 1.42i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.368 + 0.637i)5-s + (−0.499 + 0.866i)7-s + 0.353·8-s + (−0.260 − 0.450i)10-s + (0.150 + 0.261i)11-s + 1.38·13-s + (−0.353 − 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + (0.647 − 1.12i)19-s + 0.368·20-s − 0.213·22-s + (0.171 − 0.297i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.038638122\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.038638122\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.32 - 2.29i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.822 - 1.42i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 + 4.88i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.822 + 1.42i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-1.35 + 2.34i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.822 - 1.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.32 - 4.02i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.14 - 12.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.96 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 + (0.177 + 0.306i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.32 + 2.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 + (-3.29 + 5.70i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.860134939068255867611948654290, −8.803035391948845792907362712696, −8.528807695310513277442784922422, −7.37746229691133074312618428903, −6.70050936414774951729903961038, −5.95840829715130355860913176446, −5.17160470064471362199163279167, −3.79846787632667436640990619115, −3.01999055556669116457930318444, −1.46950536038448675635032126872,
0.53140231888941704425220584272, 1.53286420911197289120146696577, 3.32662839149176983205271139755, 3.70738455813615316773875832305, 4.85008144723029673143917371575, 5.87888907312900173231945641792, 6.95682920471338390867479368088, 7.83612875649662461853014613254, 8.435117197362705521143937101050, 9.420977811665216987970842626145