| L(s) = 1 | − 5-s − 11-s − 6·13-s + 2·17-s − 8·19-s + 8·23-s + 25-s − 6·29-s − 2·37-s − 6·41-s + 8·43-s + 8·47-s − 2·53-s + 55-s − 6·59-s + 8·61-s + 6·65-s + 10·67-s − 8·71-s − 14·73-s + 10·79-s − 18·83-s − 2·85-s + 14·89-s + 8·95-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s − 0.274·53-s + 0.134·55-s − 0.781·59-s + 1.02·61-s + 0.744·65-s + 1.22·67-s − 0.949·71-s − 1.63·73-s + 1.12·79-s − 1.97·83-s − 0.216·85-s + 1.48·89-s + 0.820·95-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
| 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
−14.25313219807174, −13.33191098246454, −13.00942891536756, −12.57551447291631, −12.15222202383727, −11.62981229619056, −11.01349796147080, −10.60221786421205, −10.20177766688694, −9.516714775162448, −9.056405281689939, −8.595152644739903, −7.966329619933107, −7.409176697943531, −7.099571240550861, −6.566508456378802, −5.751517295948228, −5.313034968764035, −4.670981029289811, −4.302785238961429, −3.565375306880712, −2.890942310026862, −2.367199763978805, −1.738470880027815, −0.7004744785182929, 0,
0.7004744785182929, 1.738470880027815, 2.367199763978805, 2.890942310026862, 3.565375306880712, 4.302785238961429, 4.670981029289811, 5.313034968764035, 5.751517295948228, 6.566508456378802, 7.099571240550861, 7.409176697943531, 7.966329619933107, 8.595152644739903, 9.056405281689939, 9.516714775162448, 10.20177766688694, 10.60221786421205, 11.01349796147080, 11.62981229619056, 12.15222202383727, 12.57551447291631, 13.00942891536756, 13.33191098246454, 14.25313219807174
