| L(s) = 1 | + 2·5-s + 4·11-s − 2·13-s − 6·17-s + 4·19-s − 25-s − 2·29-s − 6·37-s + 2·41-s + 4·43-s + 6·53-s + 8·55-s − 12·59-s − 2·61-s − 4·65-s − 4·67-s + 6·73-s − 16·79-s + 12·83-s − 12·85-s − 14·89-s + 8·95-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 0.986·37-s + 0.312·41-s + 0.609·43-s + 0.824·53-s + 1.07·55-s − 1.56·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.702·73-s − 1.80·79-s + 1.31·83-s − 1.30·85-s − 1.48·89-s + 0.820·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.40474512163364, −14.98037475720863, −14.24790277657576, −13.87992466134726, −13.53530837751726, −12.85210438827938, −12.22031343993403, −11.82367352132947, −11.10579274093296, −10.73460473596358, −9.829117951299122, −9.593389441506788, −9.004184177491189, −8.588553744868810, −7.674468523920436, −7.056153072959076, −6.636835957682069, −5.921326102927595, −5.493639945745523, −4.644767471442539, −4.151911355962541, −3.344657252745787, −2.532160496641370, −1.869967830632051, −1.214852225393285, 0,
1.214852225393285, 1.869967830632051, 2.532160496641370, 3.344657252745787, 4.151911355962541, 4.644767471442539, 5.493639945745523, 5.921326102927595, 6.636835957682069, 7.056153072959076, 7.674468523920436, 8.588553744868810, 9.004184177491189, 9.593389441506788, 9.829117951299122, 10.73460473596358, 11.10579274093296, 11.82367352132947, 12.22031343993403, 12.85210438827938, 13.53530837751726, 13.87992466134726, 14.24790277657576, 14.98037475720863, 15.40474512163364
