L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.866 − 0.5i)3-s − 1.00i·4-s + (−1.30 + 0.349i)5-s + (0.965 − 0.258i)6-s + (−1.83 − 1.90i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (0.675 − 1.16i)10-s + (2.57 − 0.689i)11-s + (−0.500 + 0.866i)12-s + (−3.33 + 1.36i)13-s + (2.64 + 0.0509i)14-s + (1.30 + 0.349i)15-s − 1.00·16-s + 5.24·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (−0.583 + 0.156i)5-s + (0.394 − 0.105i)6-s + (−0.693 − 0.720i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (0.213 − 0.369i)10-s + (0.775 − 0.207i)11-s + (−0.144 + 0.250i)12-s + (−0.925 + 0.379i)13-s + (0.706 + 0.0136i)14-s + (0.336 + 0.0902i)15-s − 0.250·16-s + 1.27·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.331544 + 0.412863i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.331544 + 0.412863i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (1.83 + 1.90i)T \) |
| 13 | \( 1 + (3.33 - 1.36i)T \) |
good | 5 | \( 1 + (1.30 - 0.349i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.57 + 0.689i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 + (1.78 - 6.66i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 0.262iT - 23T^{2} \) |
| 29 | \( 1 + (-3.07 - 5.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.112 - 0.421i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.94 - 1.94i)T + 37iT^{2} \) |
| 41 | \( 1 + (2.81 - 10.5i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.74 - 3.89i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.48 + 5.55i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.878 - 1.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.09 - 6.09i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.62 - 4.40i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.68 + 10.0i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.08 - 4.04i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-4.86 - 1.30i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.12 + 5.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.32 - 9.32i)T + 83iT^{2} \) |
| 89 | \( 1 + (-4.38 + 4.38i)T - 89iT^{2} \) |
| 97 | \( 1 + (18.5 - 4.97i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.90761186268877778102358709326, −10.07626669481246292075531885443, −9.440159910078327785461843214306, −8.091753665127692874349788908713, −7.46503225545660306903153017950, −6.62223681735868478759953898080, −5.83753320785919765116105457941, −4.49161297564806428567776670250, −3.36762140655425861335971770307, −1.29269480041761877403102107237,
0.42770223745161770309365823612, 2.47143656568227808358829674658, 3.67452962928099683564883713235, 4.77288991421136306073950701784, 5.94710121169077358525745778187, 7.02334756305371005631016436514, 7.962696207120044991399038177327, 9.102878816642725670181063734636, 9.627885165980127046223984791630, 10.50235436733119770389747128444