| L(s) = 1 | + (3.27 + 1.76i)2-s + (−4.06 + 3.88i)3-s + (5.39 + 8.17i)4-s + (−4.01 + 4.20i)5-s + (−20.1 + 5.55i)6-s + (11.7 − 6.33i)7-s + (1.93 + 21.4i)8-s + (1.01 − 22.5i)9-s + (−20.5 + 6.67i)10-s + (0.0465 + 0.343i)11-s + (−53.6 − 12.2i)12-s + (−2.68 + 6.28i)13-s + 49.6·14-s − 32.6i·15-s + (−16.0 + 37.5i)16-s + (−0.973 − 1.11i)17-s + ⋯ |
| L(s) = 1 | + (1.63 + 0.880i)2-s + (−1.35 + 1.29i)3-s + (1.34 + 2.04i)4-s + (−0.803 + 0.840i)5-s + (−3.35 + 0.925i)6-s + (1.68 − 0.905i)7-s + (0.241 + 2.68i)8-s + (0.112 − 2.50i)9-s + (−2.05 + 0.667i)10-s + (0.00422 + 0.0312i)11-s + (−4.47 − 1.02i)12-s + (−0.206 + 0.483i)13-s + 3.54·14-s − 2.17i·15-s + (−1.00 + 2.34i)16-s + (−0.0572 − 0.0655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0522801 + 2.46236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0522801 + 2.46236i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 211 | \( 1 + (208. + 35.4i)T \) |
| good | 2 | \( 1 + (-3.27 - 1.76i)T + (2.20 + 3.33i)T^{2} \) |
| 3 | \( 1 + (4.06 - 3.88i)T + (0.403 - 8.99i)T^{2} \) |
| 5 | \( 1 + (4.01 - 4.20i)T + (-1.12 - 24.9i)T^{2} \) |
| 7 | \( 1 + (-11.7 + 6.33i)T + (26.9 - 40.8i)T^{2} \) |
| 11 | \( 1 + (-0.0465 - 0.343i)T + (-116. + 32.1i)T^{2} \) |
| 13 | \( 1 + (2.68 - 6.28i)T + (-116. - 122. i)T^{2} \) |
| 17 | \( 1 + (0.973 + 1.11i)T + (-38.7 + 286. i)T^{2} \) |
| 19 | \( 1 + (-9.30 - 6.76i)T + (111. + 343. i)T^{2} \) |
| 23 | \( 1 + (16.6 + 22.8i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-7.73 - 3.30i)T + (581. + 607. i)T^{2} \) |
| 31 | \( 1 + (-18.2 - 4.16i)T + (865. + 416. i)T^{2} \) |
| 37 | \( 1 + (0.854 - 6.30i)T + (-1.31e3 - 364. i)T^{2} \) |
| 41 | \( 1 + (13.0 - 21.8i)T + (-796. - 1.48e3i)T^{2} \) |
| 43 | \( 1 + (5.19 - 22.7i)T + (-1.66e3 - 802. i)T^{2} \) |
| 47 | \( 1 + (-3.22 - 71.8i)T + (-2.20e3 + 198. i)T^{2} \) |
| 53 | \( 1 + (-37.5 + 56.8i)T + (-1.10e3 - 2.58e3i)T^{2} \) |
| 59 | \( 1 + (83.0 + 49.6i)T + (1.64e3 + 3.06e3i)T^{2} \) |
| 61 | \( 1 + (-58.5 + 19.0i)T + (3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-20.1 + 41.8i)T + (-2.79e3 - 3.50e3i)T^{2} \) |
| 71 | \( 1 + (12.3 + 37.8i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-2.68 + 3.36i)T + (-1.18e3 - 5.19e3i)T^{2} \) |
| 79 | \( 1 + (35.0 + 3.15i)T + (6.14e3 + 1.11e3i)T^{2} \) |
| 83 | \( 1 + (16.7 - 51.5i)T + (-5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-15.2 + 84.0i)T + (-7.41e3 - 2.78e3i)T^{2} \) |
| 97 | \( 1 + (-54.4 + 144. i)T + (-7.08e3 - 6.19e3i)T^{2} \) |
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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
−12.36737902654197194460389318215, −11.46995429473113486287728957117, −11.26728802332961765926743750080, −10.24631906946897448989724496211, −8.094881070880850155422516351540, −7.10653166078759580238321733595, −6.17346452647733217495620842182, −4.86816413423391768078031846965, −4.48112523928121340895781703059, −3.54449934271765505745097605200,
1.07353231970903985717416296535, 2.18204935950989040068046390427, 4.43923069286108241885518776266, 5.27504403572348681806074397958, 5.75545435701982877924784725384, 7.29449856881468135388341262491, 8.311442725544970369532342511422, 10.55996563545062824224354455476, 11.46779549465268461416954939782, 11.99759488020762434842508204656
