| L(s) = 1 | + 2·5-s + 7-s − 6·13-s − 17-s − 8·23-s − 25-s + 6·29-s + 8·31-s + 2·35-s + 10·37-s + 6·41-s − 12·43-s + 49-s + 10·53-s − 8·59-s + 6·61-s − 12·65-s − 12·67-s − 6·73-s + 8·79-s + 16·83-s − 2·85-s − 2·89-s − 6·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.66·13-s − 0.242·17-s − 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.338·35-s + 1.64·37-s + 0.937·41-s − 1.82·43-s + 1/7·49-s + 1.37·53-s − 1.04·59-s + 0.768·61-s − 1.48·65-s − 1.46·67-s − 0.702·73-s + 0.900·79-s + 1.75·83-s − 0.216·85-s − 0.211·89-s − 0.628·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.241241782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241241782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.86841930541051, −15.24444764881809, −14.63579114167410, −14.31438848815575, −13.53109328452391, −13.39598996081250, −12.44183497283233, −11.86805616973449, −11.72356173550880, −10.59603146845754, −10.18845752099148, −9.720560597206966, −9.277898450799781, −8.324486706081285, −7.932963007000844, −7.268764060721294, −6.436145359434298, −6.050879233584920, −5.272573726375637, −4.627840979479585, −4.157062403146158, −2.930121380130786, −2.371381838629969, −1.768739796019141, −0.6184065769108444,
0.6184065769108444, 1.768739796019141, 2.371381838629969, 2.930121380130786, 4.157062403146158, 4.627840979479585, 5.272573726375637, 6.050879233584920, 6.436145359434298, 7.268764060721294, 7.932963007000844, 8.324486706081285, 9.277898450799781, 9.720560597206966, 10.18845752099148, 10.59603146845754, 11.72356173550880, 11.86805616973449, 12.44183497283233, 13.39598996081250, 13.53109328452391, 14.31438848815575, 14.63579114167410, 15.24444764881809, 15.86841930541051
