| L(s) = 1 | − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s + 2·14-s − 16-s − 5·17-s + 6·19-s − 20-s − 2·23-s − 4·25-s + 2·28-s + 9·29-s + 2·31-s − 5·32-s + 5·34-s − 2·35-s + 3·37-s − 6·38-s + 3·40-s + 5·41-s + 2·46-s + 2·47-s − 3·49-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s + 0.534·14-s − 1/4·16-s − 1.21·17-s + 1.37·19-s − 0.223·20-s − 0.417·23-s − 4/5·25-s + 0.377·28-s + 1.67·29-s + 0.359·31-s − 0.883·32-s + 0.857·34-s − 0.338·35-s + 0.493·37-s − 0.973·38-s + 0.474·40-s + 0.780·41-s + 0.294·46-s + 0.291·47-s − 3/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.49761418179542, −12.96501269136622, −12.46034501629307, −12.04219260536040, −11.25111070446307, −11.07937539356782, −10.25200177196788, −9.961391206997401, −9.616807875678070, −9.196224026729287, −8.712855608407382, −8.178826536945973, −7.723568458628547, −7.221705220069811, −6.541198047686230, −6.238812155007882, −5.607328841404055, −4.953306040072954, −4.551283643949390, −3.949740066798557, −3.342530309359071, −2.677881983951468, −2.115136644819619, −1.323887896675613, −0.7374819379136046, 0,
0.7374819379136046, 1.323887896675613, 2.115136644819619, 2.677881983951468, 3.342530309359071, 3.949740066798557, 4.551283643949390, 4.953306040072954, 5.607328841404055, 6.238812155007882, 6.541198047686230, 7.221705220069811, 7.723568458628547, 8.178826536945973, 8.712855608407382, 9.196224026729287, 9.616807875678070, 9.961391206997401, 10.25200177196788, 11.07937539356782, 11.25111070446307, 12.04219260536040, 12.46034501629307, 12.96501269136622, 13.49761418179542
