| L(s) = 1 | − 7-s − 4·11-s − 2·13-s − 6·17-s − 8·19-s − 6·29-s − 8·31-s + 2·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s − 6·61-s − 4·67-s − 8·71-s − 10·73-s + 4·77-s − 16·79-s − 8·83-s + 6·89-s + 2·91-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.768·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.80·79-s − 0.878·83-s + 0.635·89-s + 0.209·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.89960522179933, −15.19652847474308, −15.00558375429185, −14.43133483960224, −13.46701886196311, −13.21446964501041, −12.81190277459308, −12.27805406503613, −11.46057209468137, −10.94850076625075, −10.48242312986782, −10.04891498422979, −9.161860134823984, −8.824345973958107, −8.247480300114612, −7.333373742027227, −7.182427397259113, −6.277704789907098, −5.781665312500630, −5.087084129118901, −4.369628947821559, −3.910770932654989, −2.878390043197332, −2.337733081975300, −1.716947658537234, 0, 0,
1.716947658537234, 2.337733081975300, 2.878390043197332, 3.910770932654989, 4.369628947821559, 5.087084129118901, 5.781665312500630, 6.277704789907098, 7.182427397259113, 7.333373742027227, 8.247480300114612, 8.824345973958107, 9.161860134823984, 10.04891498422979, 10.48242312986782, 10.94850076625075, 11.46057209468137, 12.27805406503613, 12.81190277459308, 13.21446964501041, 13.46701886196311, 14.43133483960224, 15.00558375429185, 15.19652847474308, 15.89960522179933
