L(s) = 1 | + 7-s − 2·11-s − 2·13-s − 4·17-s + 4·19-s − 6·23-s − 5·25-s + 2·29-s − 6·37-s − 8·41-s + 8·43-s − 4·47-s + 49-s + 6·53-s − 14·61-s − 4·67-s − 2·71-s − 2·73-s − 2·77-s − 4·79-s + 12·83-s − 2·91-s + 6·97-s − 12·101-s + 8·103-s + 6·107-s − 18·109-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.917·19-s − 1.25·23-s − 25-s + 0.371·29-s − 0.986·37-s − 1.24·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.79·61-s − 0.488·67-s − 0.237·71-s − 0.234·73-s − 0.227·77-s − 0.450·79-s + 1.31·83-s − 0.209·91-s + 0.609·97-s − 1.19·101-s + 0.788·103-s + 0.580·107-s − 1.72·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.728465219026174325197401380467, −7.908788854799808135518860558091, −7.35099727767244243949436898839, −6.37197187863691936186639353981, −5.49391588685950886857050871947, −4.74120936441215238109926770183, −3.82735621857672251034800538625, −2.67528968157297083403654922205, −1.72748916226402630622539746394, 0,
1.72748916226402630622539746394, 2.67528968157297083403654922205, 3.82735621857672251034800538625, 4.74120936441215238109926770183, 5.49391588685950886857050871947, 6.37197187863691936186639353981, 7.35099727767244243949436898839, 7.908788854799808135518860558091, 8.728465219026174325197401380467