| L(s) = 1 | + (−0.123 − 1.52i)2-s + (0.845 − 0.534i)3-s + (−0.350 + 0.0570i)4-s + (−0.731 − 0.826i)5-s + (−0.922 − 1.22i)6-s + (−0.602 + 0.578i)7-s + (−0.604 − 2.45i)8-s + (0.428 − 0.903i)9-s + (−1.17 + 1.22i)10-s + (5.24 − 2.49i)11-s + (−0.266 + 0.235i)12-s + (3.38 + 1.23i)13-s + (0.959 + 0.850i)14-s + (−1.05 − 0.307i)15-s + (−4.34 + 1.45i)16-s + (−4.41 − 4.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.0873 − 1.08i)2-s + (0.487 − 0.308i)3-s + (−0.175 + 0.0285i)4-s + (−0.327 − 0.369i)5-s + (−0.376 − 0.500i)6-s + (−0.227 + 0.218i)7-s + (−0.213 − 0.866i)8-s + (0.142 − 0.301i)9-s + (−0.371 + 0.386i)10-s + (1.58 − 0.750i)11-s + (−0.0768 + 0.0680i)12-s + (0.939 + 0.343i)13-s + (0.256 + 0.227i)14-s + (−0.273 − 0.0792i)15-s + (−1.08 + 0.362i)16-s + (−1.07 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.514733 - 1.56104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.514733 - 1.56104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.845 + 0.534i)T \) |
| 13 | \( 1 + (-3.38 - 1.23i)T \) |
| good | 2 | \( 1 + (0.123 + 1.52i)T + (-1.97 + 0.320i)T^{2} \) |
| 5 | \( 1 + (0.731 + 0.826i)T + (-0.602 + 4.96i)T^{2} \) |
| 7 | \( 1 + (0.602 - 0.578i)T + (0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-5.24 + 2.49i)T + (6.95 - 8.52i)T^{2} \) |
| 17 | \( 1 + (4.41 + 4.59i)T + (-0.684 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.82 - 2.20i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.20 - 5.55i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.71 + 0.138i)T + (28.6 - 4.65i)T^{2} \) |
| 31 | \( 1 + (0.273 - 0.0331i)T + (30.0 - 7.41i)T^{2} \) |
| 37 | \( 1 + (-1.31 + 3.07i)T + (-25.6 - 26.6i)T^{2} \) |
| 41 | \( 1 + (-3.85 - 6.09i)T + (-17.5 + 37.0i)T^{2} \) |
| 43 | \( 1 + (2.75 - 1.17i)T + (29.7 - 31.0i)T^{2} \) |
| 47 | \( 1 + (-10.0 + 3.79i)T + (35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (4.80 - 1.18i)T + (46.9 - 24.6i)T^{2} \) |
| 59 | \( 1 + (-3.08 + 9.23i)T + (-47.1 - 35.4i)T^{2} \) |
| 61 | \( 1 + (0.577 + 1.99i)T + (-51.5 + 32.6i)T^{2} \) |
| 67 | \( 1 + (0.924 - 5.68i)T + (-63.5 - 21.2i)T^{2} \) |
| 71 | \( 1 + (-4.69 + 0.189i)T + (70.7 - 5.71i)T^{2} \) |
| 73 | \( 1 + (-7.80 - 5.39i)T + (25.8 + 68.2i)T^{2} \) |
| 79 | \( 1 + (-4.32 - 11.4i)T + (-59.1 + 52.3i)T^{2} \) |
| 83 | \( 1 + (0.413 + 0.788i)T + (-47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-9.21 - 5.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 2.73i)T + (89.2 + 38.0i)T^{2} \) |
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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
−10.81450058607858719148169444208, −9.477646909464230495761467952327, −9.082618462406497663518026407519, −8.154767891626248383482427544427, −6.74191602879502545966464858147, −6.17254250307753978874866043384, −4.21437049185244699965862656708, −3.57178942848606837583532979359, −2.25925866722772654166318671694, −1.02094192050254084207606941982,
2.11295392288527036543124007232, 3.71528759900942507435945337402, 4.52660562181707919953320473046, 6.20247514319209656212467633182, 6.61148681446091201797285366780, 7.55503238949748111426791686705, 8.646798677385639165416666311173, 8.994821607649803524894295043705, 10.41779887752402612453066399937, 11.07575777595604891028114196091
