L(s) = 1 | − 5-s − 11-s − 6·13-s − 6·17-s − 8·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s + 10·41-s + 6·53-s + 55-s + 2·59-s − 8·61-s + 6·65-s + 2·67-s − 8·71-s − 14·73-s + 10·79-s + 6·83-s + 6·85-s + 6·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s − 1.45·17-s − 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 1.56·41-s + 0.824·53-s + 0.134·55-s + 0.260·59-s − 1.02·61-s + 0.744·65-s + 0.244·67-s − 0.949·71-s − 1.63·73-s + 1.12·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.76666000829100, −12.07390046535660, −11.92916628709336, −11.36897637354371, −10.85201440001951, −10.48824041430698, −9.995729466447198, −9.470346112728536, −9.121830999617610, −8.650111564256519, −7.992285444223024, −7.588694107798674, −7.377528246559554, −6.737921959202364, −6.224790483865760, −5.756491447193908, −5.153989568458323, −4.679043781169356, −4.148809300510450, −3.935444547628414, −3.029782315015726, −2.440602769676609, −2.239357870371679, −1.481792136818798, −0.4844507934663634, 0,
0.4844507934663634, 1.481792136818798, 2.239357870371679, 2.440602769676609, 3.029782315015726, 3.935444547628414, 4.148809300510450, 4.679043781169356, 5.153989568458323, 5.756491447193908, 6.224790483865760, 6.737921959202364, 7.377528246559554, 7.588694107798674, 7.992285444223024, 8.650111564256519, 9.121830999617610, 9.470346112728536, 9.995729466447198, 10.48824041430698, 10.85201440001951, 11.36897637354371, 11.92916628709336, 12.07390046535660, 12.76666000829100