L(s) = 1 | + (0.866 + 0.5i)2-s + 3-s + (0.499 + 0.866i)4-s + (−0.0451 + 0.0260i)5-s + (0.866 + 0.5i)6-s + (1.51 − 2.16i)7-s + 0.999i·8-s + 9-s − 0.0521·10-s + 4.38i·11-s + (0.499 + 0.866i)12-s + (1.30 − 3.35i)13-s + (2.39 − 1.11i)14-s + (−0.0451 + 0.0260i)15-s + (−0.5 + 0.866i)16-s + (2.22 + 3.85i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + 0.577·3-s + (0.249 + 0.433i)4-s + (−0.0201 + 0.0116i)5-s + (0.353 + 0.204i)6-s + (0.573 − 0.819i)7-s + 0.353i·8-s + 0.333·9-s − 0.0164·10-s + 1.32i·11-s + (0.144 + 0.249i)12-s + (0.362 − 0.931i)13-s + (0.640 − 0.298i)14-s + (−0.0116 + 0.00672i)15-s + (−0.125 + 0.216i)16-s + (0.539 + 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.51858 + 0.659634i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.51858 + 0.659634i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + (-1.51 + 2.16i)T \) |
| 13 | \( 1 + (-1.30 + 3.35i)T \) |
good | 5 | \( 1 + (0.0451 - 0.0260i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 4.38iT - 11T^{2} \) |
| 17 | \( 1 + (-2.22 - 3.85i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 2.15iT - 19T^{2} \) |
| 23 | \( 1 + (-2.35 + 4.07i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.37 + 2.38i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.373 + 0.215i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.92 + 2.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0861 + 0.0497i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.18 - 8.98i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0347 - 0.0200i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.481 - 0.834i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.22 - 2.44i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 4.08T + 61T^{2} \) |
| 67 | \( 1 + 6.65iT - 67T^{2} \) |
| 71 | \( 1 + (4.23 + 2.44i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.34 + 0.774i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.28 + 14.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + (-5.77 - 3.33i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.77 - 2.17i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.76950112349325498338803339135, −10.13401857900644148266688940271, −8.969742944227534950570274620755, −7.85961287046265833925256748699, −7.46507390799984519447145866928, −6.36565791523419245332968793594, −5.08253307363845265146404700358, −4.24438250631053864680784232435, −3.22049909633552163672989663774, −1.71344888753611276429155313929,
1.59174803333995112449078696254, 2.88051173691563139356338375753, 3.82086525619526578146911635401, 5.10299823884893532987684057382, 5.90050297851056847933022205812, 7.06524648315727991795937131820, 8.258525461184112435118650068591, 8.911407319642477997303015412340, 9.827242886220920244653320273091, 10.97834232255484033288237830037