L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + 0.692i·5-s + (−0.866 − 0.499i)6-s + (−0.309 − 0.178i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.346 + 0.599i)10-s + (2.54 − 1.46i)11-s − 0.999·12-s − 0.356·14-s + (0.599 − 0.346i)15-s + (−0.5 − 0.866i)16-s + (3.35 − 5.81i)17-s + 0.999i·18-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + 0.309i·5-s + (−0.353 − 0.204i)6-s + (−0.116 − 0.0674i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.109 + 0.189i)10-s + (0.767 − 0.443i)11-s − 0.288·12-s − 0.0953·14-s + (0.154 − 0.0893i)15-s + (−0.125 − 0.216i)16-s + (0.814 − 1.41i)17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.934649602\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.934649602\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.692iT - 5T^{2} \) |
| 7 | \( 1 + (0.309 + 0.178i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.54 + 1.46i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.35 + 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (6.24 + 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.91 + 6.78i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.76iT - 31T^{2} \) |
| 37 | \( 1 + (-8.74 + 5.04i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.23 - 2.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.29 + 5.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.98iT - 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 + (1.42 + 0.821i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (11.7 - 6.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.89 - 3.40i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 3.18iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.8iT - 83T^{2} \) |
| 89 | \( 1 + (-0.343 + 0.198i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.361 + 0.208i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.753503253713921096550081863417, −9.039804394197761634152093969002, −7.86622886641398385548390680103, −6.92765230826597513144588245818, −6.33731580915258787752513315859, −5.37655355737591988169644319156, −4.40419015963969025082286422569, −3.27438716390902760714011463025, −2.28464771616379812472662900657, −0.77969031093146185612092570878,
1.64917370349581000121612116177, 3.27537980630373570462641092481, 4.17022992718970969253746920076, 4.87550557819395757079458930958, 6.05005869535002173521748208026, 6.44201823213364385518028296106, 7.68881207461435998829972454189, 8.536529639625709044997596799680, 9.317370204085383840916245724796, 10.34107545047970774659629484970