L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (1.69 − 1.23i)5-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 2.10·10-s + (3.19 + 0.871i)11-s − 12-s + (4.47 + 3.25i)13-s + (0.309 − 0.951i)14-s + (0.649 + 1.99i)15-s + (−0.809 + 0.587i)16-s + (−1.5 + 1.08i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.760 − 0.552i)5-s + (−0.330 + 0.239i)6-s + (−0.116 − 0.359i)7-s + (−0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.664·10-s + (0.964 + 0.262i)11-s − 0.288·12-s + (1.24 + 0.902i)13-s + (0.0825 − 0.254i)14-s + (0.167 + 0.515i)15-s + (−0.202 + 0.146i)16-s + (−0.363 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83928 + 1.02208i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83928 + 1.02208i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (-3.19 - 0.871i)T \) |
good | 5 | \( 1 + (-1.69 + 1.23i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-4.47 - 3.25i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.5 - 1.08i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.283 + 0.871i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + (-0.675 - 2.07i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (8.18 + 5.94i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.11 - 3.43i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.45 + 10.6i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.17T + 43T^{2} \) |
| 47 | \( 1 + (-1.29 + 3.97i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.76 - 4.18i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.67 - 8.23i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.11 + 2.98i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + (1.16 - 0.845i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.05 + 12.4i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (14.3 + 10.4i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.34 + 6.79i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 3.87T + 89T^{2} \) |
| 97 | \( 1 + (4.62 + 3.36i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.29318010763014506216721611073, −10.31709982418338105993139005392, −9.132566309346361376486477769390, −8.843968273913171154540949192339, −7.27838600128063090797153762516, −6.30577695060055915489911474259, −5.58998766448610914857797864442, −4.37167739737226637882980253298, −3.68910344414662785944138905605, −1.73753360116089454375486257347,
1.41170084791829956646693722381, 2.72202602981747947463494175394, 3.83118178540950920374522978699, 5.43778037831583525539011391476, 6.14547288449986196067186254221, 6.81892926293019357647237495176, 8.226105976809262264430743165550, 9.225780012964512611217046236825, 10.22788337347187669682148418464, 11.07506679002424272040609608578