| L(s) = 1 | + 3-s + 2·5-s − 7-s + 9-s + 6·13-s + 2·15-s − 2·17-s + 4·19-s − 21-s − 25-s + 27-s − 2·29-s + 8·31-s − 2·35-s − 6·37-s + 6·39-s − 10·41-s − 4·43-s + 2·45-s − 8·47-s + 49-s − 2·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.338·35-s − 0.986·37-s + 0.960·39-s − 1.56·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.58986133450993, −13.13133955995909, −12.82249704107228, −12.01488016097310, −11.62631764915462, −11.14078608610893, −10.45498765835337, −10.15454890730323, −9.646785798512175, −9.216675178336634, −8.699338713961317, −8.304821697827577, −7.829143268829092, −7.098140172295971, −6.550419285285313, −6.240240144163529, −5.736877986963269, −5.041591296652738, −4.614166596635418, −3.741452321766847, −3.369198067647187, −2.940957130596175, −2.074022019461224, −1.606705114336533, −1.076040946316660, 0,
1.076040946316660, 1.606705114336533, 2.074022019461224, 2.940957130596175, 3.369198067647187, 3.741452321766847, 4.614166596635418, 5.041591296652738, 5.736877986963269, 6.240240144163529, 6.550419285285313, 7.098140172295971, 7.829143268829092, 8.304821697827577, 8.699338713961317, 9.216675178336634, 9.646785798512175, 10.15454890730323, 10.45498765835337, 11.14078608610893, 11.62631764915462, 12.01488016097310, 12.82249704107228, 13.13133955995909, 13.58986133450993
