| L(s) = 1 | + 3-s − 5·7-s + 9-s + 13-s − 17-s − 6·19-s − 5·21-s − 4·23-s − 5·25-s + 27-s + 3·29-s − 9·31-s + 37-s + 39-s + 6·41-s − 3·43-s + 47-s + 18·49-s − 51-s + 2·53-s − 6·57-s − 7·59-s + 4·61-s − 5·63-s − 2·67-s − 4·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.277·13-s − 0.242·17-s − 1.37·19-s − 1.09·21-s − 0.834·23-s − 25-s + 0.192·27-s + 0.557·29-s − 1.61·31-s + 0.164·37-s + 0.160·39-s + 0.937·41-s − 0.457·43-s + 0.145·47-s + 18/7·49-s − 0.140·51-s + 0.274·53-s − 0.794·57-s − 0.911·59-s + 0.512·61-s − 0.629·63-s − 0.244·67-s − 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320892 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.93989844302772, −12.66982007031465, −11.95980567515224, −11.64500808325094, −10.85608267700391, −10.42196329959175, −10.16354533166047, −9.593329857301495, −9.086331482421966, −8.971192865592366, −8.293924825985034, −7.737641131791404, −7.327962840175278, −6.704515992215826, −6.346551425771763, −5.962531491465874, −5.496197652137974, −4.617729206421582, −4.024696685908197, −3.789816734337057, −3.237112158027357, −2.640012594037948, −2.173237994423510, −1.595205953006060, −0.5610007280523541, 0,
0.5610007280523541, 1.595205953006060, 2.173237994423510, 2.640012594037948, 3.237112158027357, 3.789816734337057, 4.024696685908197, 4.617729206421582, 5.496197652137974, 5.962531491465874, 6.346551425771763, 6.704515992215826, 7.327962840175278, 7.737641131791404, 8.293924825985034, 8.971192865592366, 9.086331482421966, 9.593329857301495, 10.16354533166047, 10.42196329959175, 10.85608267700391, 11.64500808325094, 11.95980567515224, 12.66982007031465, 12.93989844302772
