| L(s) = 1 | − 3-s − 3·5-s + 7-s − 2·9-s + 2·11-s − 2·13-s + 3·15-s − 17-s + 19-s − 21-s + 8·23-s + 4·25-s + 5·27-s − 29-s − 2·33-s − 3·35-s − 2·37-s + 2·39-s − 7·41-s + 8·43-s + 6·45-s + 8·47-s − 6·49-s + 51-s + 3·53-s − 6·55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s + 0.774·15-s − 0.242·17-s + 0.229·19-s − 0.218·21-s + 1.66·23-s + 4/5·25-s + 0.962·27-s − 0.185·29-s − 0.348·33-s − 0.507·35-s − 0.328·37-s + 0.320·39-s − 1.09·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s − 0.809·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.095575659078958781344328488228, −7.23427701735385486286027819341, −6.82930689171494415309635766595, −5.75281845521053839460736579453, −5.02872490792614591051309557729, −4.34515175995065085137023566002, −3.49127357774111825372602476656, −2.62377678048319806853132040337, −1.11782411776987817886590080677, 0,
1.11782411776987817886590080677, 2.62377678048319806853132040337, 3.49127357774111825372602476656, 4.34515175995065085137023566002, 5.02872490792614591051309557729, 5.75281845521053839460736579453, 6.82930689171494415309635766595, 7.23427701735385486286027819341, 8.095575659078958781344328488228
