| L(s) = 1 | − 2·3-s + 5-s + 2·7-s + 9-s − 2·13-s − 2·15-s + 6·17-s + 4·19-s − 4·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s + 2·35-s + 4·39-s + 6·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 12·51-s − 6·53-s − 8·57-s − 12·59-s − 2·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.338·35-s + 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.05·57-s − 1.56·59-s − 0.256·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.73293445076612, −14.81744213155411, −14.37791385863995, −14.11105998216809, −13.39145829010546, −12.63642850164527, −12.12489161474588, −11.88683772024898, −11.18358540979484, −10.81280921120883, −10.12470756691435, −9.668262685412400, −9.164948245054838, −8.188652690327247, −7.723451484769091, −7.307795757093273, −6.335549662424134, −5.884594085062029, −5.474254310250073, −4.876630534318188, −4.319618768869153, −3.363895032105446, −2.626816422721340, −1.657501656359819, −1.047042524082576, 0,
1.047042524082576, 1.657501656359819, 2.626816422721340, 3.363895032105446, 4.319618768869153, 4.876630534318188, 5.474254310250073, 5.884594085062029, 6.335549662424134, 7.307795757093273, 7.723451484769091, 8.188652690327247, 9.164948245054838, 9.668262685412400, 10.12470756691435, 10.81280921120883, 11.18358540979484, 11.88683772024898, 12.12489161474588, 12.63642850164527, 13.39145829010546, 14.11105998216809, 14.37791385863995, 14.81744213155411, 15.73293445076612
