L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s + 13-s + 6·17-s − 8·19-s + 2·21-s − 9·23-s − 27-s + 4·29-s + 5·31-s + 33-s − 2·37-s − 39-s − 11·41-s − 43-s − 11·47-s − 3·49-s − 6·51-s − 3·53-s + 8·57-s + 5·59-s + 14·61-s − 2·63-s − 16·67-s + 9·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.45·17-s − 1.83·19-s + 0.436·21-s − 1.87·23-s − 0.192·27-s + 0.742·29-s + 0.898·31-s + 0.174·33-s − 0.328·37-s − 0.160·39-s − 1.71·41-s − 0.152·43-s − 1.60·47-s − 3/7·49-s − 0.840·51-s − 0.412·53-s + 1.05·57-s + 0.650·59-s + 1.79·61-s − 0.251·63-s − 1.95·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.24446088815008, −12.76165464871359, −12.19177835208114, −12.02939570209291, −11.48396127896055, −10.90179206159862, −10.14335185498544, −10.10056141057384, −9.920566336514560, −8.950725530017982, −8.407846369867486, −8.154530010276632, −7.573988390858457, −6.873972236253825, −6.396563660212718, −6.126448770068348, −5.674302761065635, −4.847890398563283, −4.634678335309211, −3.745491295669099, −3.491894203511899, −2.792816688064691, −2.018190074732220, −1.555800374547264, −0.6033333730113693, 0,
0.6033333730113693, 1.555800374547264, 2.018190074732220, 2.792816688064691, 3.491894203511899, 3.745491295669099, 4.634678335309211, 4.847890398563283, 5.674302761065635, 6.126448770068348, 6.396563660212718, 6.873972236253825, 7.573988390858457, 8.154530010276632, 8.407846369867486, 8.950725530017982, 9.920566336514560, 10.10056141057384, 10.14335185498544, 10.90179206159862, 11.48396127896055, 12.02939570209291, 12.19177835208114, 12.76165464871359, 13.24446088815008