| 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 − 2·17-s − 2·20-s − 22-s + 8·23-s − 25-s − 2·26-s − 28-s + 2·29-s − 8·31-s − 32-s + 2·34-s + 2·35-s − 2·37-s + 2·40-s − 10·41-s + 4·43-s + 44-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 − 0.485·17-s − 0.447·20-s − 0.213·22-s + 1.66·23-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 + 0.342·34-s + 0.338·35-s − 0.328·37-s + 0.316·40-s − 1.56·41-s + 0.609·43-s + 0.150·44-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 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.955994247194040414018789515096, −8.582739275429656645366707671354, −7.52239117864584703743055005677, −6.96355267990089995055745653875, −6.08896062740208954182977845395, −4.92364198553775270952134324473, −3.79941779708894965776977767454, −3.00404374709518806814432634087, −1.49721415431484567819975457145, 0,
1.49721415431484567819975457145, 3.00404374709518806814432634087, 3.79941779708894965776977767454, 4.92364198553775270952134324473, 6.08896062740208954182977845395, 6.96355267990089995055745653875, 7.52239117864584703743055005677, 8.582739275429656645366707671354, 8.955994247194040414018789515096
