| L(s) = 1 | − 0.540i·3-s − 2.90i·5-s + 2.60·7-s + 2.70·9-s + 2.61i·13-s − 1.56·15-s − 6.89·17-s − 1.60i·19-s − 1.40i·21-s + 4.01·23-s − 3.41·25-s − 3.08i·27-s − 1.99i·29-s + 5.18·31-s − 7.55i·35-s + ⋯ |
| L(s) = 1 | − 0.312i·3-s − 1.29i·5-s + 0.984·7-s + 0.902·9-s + 0.725i·13-s − 0.404·15-s − 1.67·17-s − 0.367i·19-s − 0.307i·21-s + 0.837·23-s − 0.683·25-s − 0.593i·27-s − 0.370i·29-s + 0.930·31-s − 1.27i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.145018913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145018913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.540iT - 3T^{2} \) |
| 5 | \( 1 + 2.90iT - 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 13 | \( 1 - 2.61iT - 13T^{2} \) |
| 17 | \( 1 + 6.89T + 17T^{2} \) |
| 19 | \( 1 + 1.60iT - 19T^{2} \) |
| 23 | \( 1 - 4.01T + 23T^{2} \) |
| 29 | \( 1 + 1.99iT - 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 8.58iT - 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 1.29iT - 53T^{2} \) |
| 59 | \( 1 + 0.351iT - 59T^{2} \) |
| 61 | \( 1 + 3.08iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 3.47T + 71T^{2} \) |
| 73 | \( 1 + 1.33T + 73T^{2} \) |
| 79 | \( 1 - 4.32T + 79T^{2} \) |
| 83 | \( 1 + 15.9iT - 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 8.61T + 97T^{2} \) |
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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
−8.393568992762305858202574827940, −7.52851104089077145076808019042, −6.92584122312136918257336685748, −6.06981774831427870258557617624, −4.91996480219444093552520603173, −4.64545951421078829128223774765, −3.97151008553635904341389479707, −2.32164494305229086493011179422, −1.63545306923616809526338703094, −0.66334738858411860620795175900,
1.28743204027936239305134018848, 2.39716711639003532257263345048, 3.15956100474084994765569842297, 4.21528549168006274628991489856, 4.76842825168096123534822413027, 5.71258458071706774213012651753, 6.83685769239310709693162788545, 6.95120450150727007565184786172, 7.975354137257398452244679188064, 8.568007250756743754863884383744
