| L(s) = 1 | + 2-s + 4-s − 2·5-s − 7-s + 8-s − 2·10-s − 11-s + 2·13-s − 14-s + 16-s − 6·17-s − 8·19-s − 2·20-s − 22-s − 4·23-s − 25-s + 2·26-s − 28-s − 2·29-s + 8·31-s + 32-s − 6·34-s + 2·35-s + 6·37-s − 8·38-s − 2·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.83·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.338·35-s + 0.986·37-s − 1.29·38-s − 0.316·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.025935123728764317523729419386, −8.260375317206775288322444616854, −7.54842093789349707312948138248, −6.41309505638029156456085338806, −6.10611924951092571905001273028, −4.52496529634501691830241759811, −4.24578358603152871833997851063, −3.11903064072712609997880205813, −2.02661744514455988507183410743, 0,
2.02661744514455988507183410743, 3.11903064072712609997880205813, 4.24578358603152871833997851063, 4.52496529634501691830241759811, 6.10611924951092571905001273028, 6.41309505638029156456085338806, 7.54842093789349707312948138248, 8.260375317206775288322444616854, 9.025935123728764317523729419386
