| L(s) = 1 | − 3-s + 5-s + 9-s + 2·13-s − 15-s + 6·17-s + 4·19-s + 25-s − 27-s − 6·29-s − 8·31-s + 6·37-s − 2·39-s − 10·41-s + 4·43-s + 45-s − 8·47-s − 7·49-s − 6·51-s − 10·53-s − 4·57-s − 12·59-s − 6·61-s + 2·65-s − 4·67-s + 14·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.840·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s − 0.768·61-s + 0.248·65-s − 0.488·67-s + 1.63·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−16.49229498772640, −16.02998255520771, −15.19831145718385, −14.76525761089887, −14.07313469535291, −13.63840055994050, −12.90813318859820, −12.50046409308274, −11.88728323515565, −11.12353607602551, −10.93655134643092, −10.02451678249330, −9.556222192982442, −9.171040085774202, −8.066574334736688, −7.757300909154716, −6.957054428983156, −6.303908014146177, −5.606300012934580, −5.306387009615148, −4.460880549519880, −3.495131776663428, −3.084269269947937, −1.752810535941762, −1.276072609889477, 0,
1.276072609889477, 1.752810535941762, 3.084269269947937, 3.495131776663428, 4.460880549519880, 5.306387009615148, 5.606300012934580, 6.303908014146177, 6.957054428983156, 7.757300909154716, 8.066574334736688, 9.171040085774202, 9.556222192982442, 10.02451678249330, 10.93655134643092, 11.12353607602551, 11.88728323515565, 12.50046409308274, 12.90813318859820, 13.63840055994050, 14.07313469535291, 14.76525761089887, 15.19831145718385, 16.02998255520771, 16.49229498772640
