L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.866 − 0.5i)3-s + 1.00i·4-s + (1.41 + 0.379i)5-s + (−0.965 − 0.258i)6-s + (0.378 + 2.61i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.733 − 1.27i)10-s + (−3.64 − 0.977i)11-s + (0.500 + 0.866i)12-s + (3.50 − 0.851i)13-s + (1.58 − 2.11i)14-s + (1.41 − 0.379i)15-s − 1.00·16-s + 7.07·17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.499 − 0.288i)3-s + 0.500i·4-s + (0.633 + 0.169i)5-s + (−0.394 − 0.105i)6-s + (0.143 + 0.989i)7-s + (0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.231 − 0.401i)10-s + (−1.10 − 0.294i)11-s + (0.144 + 0.250i)12-s + (0.971 − 0.236i)13-s + (0.423 − 0.566i)14-s + (0.365 − 0.0980i)15-s − 0.250·16-s + 1.71·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51101 - 0.190660i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51101 - 0.190660i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.378 - 2.61i)T \) |
| 13 | \( 1 + (-3.50 + 0.851i)T \) |
good | 5 | \( 1 + (-1.41 - 0.379i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (3.64 + 0.977i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 + (-1.26 - 4.70i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 + (-1.51 + 2.63i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.35 - 8.79i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-3.06 + 3.06i)T - 37iT^{2} \) |
| 41 | \( 1 + (0.520 + 1.94i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.52 + 2.03i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.85 + 10.6i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.155 - 0.268i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.86 + 1.86i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.74 + 1.00i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.70 - 10.0i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.304 - 1.13i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (11.8 - 3.17i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.15 + 14.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.60 - 1.60i)T + 89iT^{2} \) |
| 97 | \( 1 + (3.84 + 1.03i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.42790021789206359471392643565, −10.04451913945717180382949500965, −8.947989384899241630400112659475, −8.225233409999222641266231770080, −7.55918783502599987957824590764, −6.02860172739838524659121963941, −5.42326039022552903415614678545, −3.54114836129561638196798179549, −2.66499557861599570857960362022, −1.47469354849919936538850742238,
1.20861013534583512397960691856, 2.83994597313608631388523218834, 4.27609518534525500302551244785, 5.32496296204707013850460404054, 6.32430531712310529647553962194, 7.59843790841210058844664628338, 7.959188216059178792621582617084, 9.177725463687302690847091296341, 9.863068389848898199023710593274, 10.52087755831545903105449938030