L(s) = 1 | + 5-s − 7-s − 6·11-s − 4·13-s − 2·17-s + 23-s + 25-s + 8·29-s − 4·31-s − 35-s − 6·37-s + 10·41-s − 6·43-s − 8·47-s + 49-s + 2·53-s − 6·55-s − 2·61-s − 4·65-s + 10·67-s + 6·71-s − 10·73-s + 6·77-s − 8·79-s − 2·85-s − 8·89-s + 4·91-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.80·11-s − 1.10·13-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.169·35-s − 0.986·37-s + 1.56·41-s − 0.914·43-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.809·55-s − 0.256·61-s − 0.496·65-s + 1.22·67-s + 0.712·71-s − 1.17·73-s + 0.683·77-s − 0.900·79-s − 0.216·85-s − 0.847·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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
−13.78091346290211, −13.25091306577153, −12.91843801379103, −12.52332007822077, −12.02627291679839, −11.36589768872255, −10.79115029720940, −10.43828564432637, −9.853402531954138, −9.706452904806888, −8.864726974008733, −8.472281678610445, −7.860368702018916, −7.404310645492762, −6.854974140992329, −6.406589755618664, −5.605880206247879, −5.301301403887047, −4.746369467948925, −4.280978068156921, −3.247544060036954, −2.892594732321199, −2.315081653525397, −1.793152300511194, −0.6833696944514281, 0,
0.6833696944514281, 1.793152300511194, 2.315081653525397, 2.892594732321199, 3.247544060036954, 4.280978068156921, 4.746369467948925, 5.301301403887047, 5.605880206247879, 6.406589755618664, 6.854974140992329, 7.404310645492762, 7.860368702018916, 8.472281678610445, 8.864726974008733, 9.706452904806888, 9.853402531954138, 10.43828564432637, 10.79115029720940, 11.36589768872255, 12.02627291679839, 12.52332007822077, 12.91843801379103, 13.25091306577153, 13.78091346290211