L(s) = 1 | − 2·7-s + 6·13-s − 5·17-s + 5·19-s + 8·23-s + 3·29-s + 6·31-s − 9·37-s + 2·41-s − 10·43-s + 7·47-s − 3·49-s − 2·53-s − 9·59-s + 67-s − 3·71-s + 14·73-s + 2·79-s − 3·83-s + 89-s − 12·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 1.66·13-s − 1.21·17-s + 1.14·19-s + 1.66·23-s + 0.557·29-s + 1.07·31-s − 1.47·37-s + 0.312·41-s − 1.52·43-s + 1.02·47-s − 3/7·49-s − 0.274·53-s − 1.17·59-s + 0.122·67-s − 0.356·71-s + 1.63·73-s + 0.225·79-s − 0.329·83-s + 0.105·89-s − 1.25·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.18620962951445, −12.73994899895699, −12.21595320690057, −11.66464439316794, −11.24686451982477, −10.81839775902849, −10.42376028376965, −9.841060070450250, −9.307638594823050, −8.926218561713393, −8.517629724900668, −8.080612740793969, −7.322326049181948, −6.856733088004553, −6.472928453000575, −6.127019003100783, −5.391540177911871, −4.942650906775057, −4.412525158700231, −3.689185442623384, −3.249847102693122, −2.920573774287297, −2.105173691868574, −1.318068283452081, −0.8974831877340575, 0,
0.8974831877340575, 1.318068283452081, 2.105173691868574, 2.920573774287297, 3.249847102693122, 3.689185442623384, 4.412525158700231, 4.942650906775057, 5.391540177911871, 6.127019003100783, 6.472928453000575, 6.856733088004553, 7.322326049181948, 8.080612740793969, 8.517629724900668, 8.926218561713393, 9.307638594823050, 9.841060070450250, 10.42376028376965, 10.81839775902849, 11.24686451982477, 11.66464439316794, 12.21595320690057, 12.73994899895699, 13.18620962951445