| L(s) = 1 | − 4·7-s + 4·11-s − 4·13-s + 6·17-s + 4·19-s + 4·23-s + 4·29-s − 4·37-s + 8·41-s + 12·47-s + 9·49-s − 2·53-s − 12·59-s − 2·61-s − 8·67-s + 8·71-s − 16·73-s − 16·77-s − 8·79-s − 8·83-s + 16·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 0.742·29-s − 0.657·37-s + 1.24·41-s + 1.75·47-s + 9/7·49-s − 0.274·53-s − 1.56·59-s − 0.256·61-s − 0.977·67-s + 0.949·71-s − 1.87·73-s − 1.82·77-s − 0.900·79-s − 0.878·83-s + 1.67·91-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.850684346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.850684346\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−16.10839354723984, −15.70722104986870, −14.92405090866785, −14.41677319956909, −13.93940343252848, −13.35715336481839, −12.45717177400896, −12.31730339438544, −11.85605423359099, −10.96739546134274, −10.22417099761109, −9.788693095017872, −9.280172451423393, −8.893422377651092, −7.823261341310853, −7.214247533262595, −6.859109284234373, −5.985344757508338, −5.604132935161850, −4.646074116970887, −3.933998167257511, −3.059036517563198, −2.860558832897748, −1.465288181117594, −0.6320773149724936,
0.6320773149724936, 1.465288181117594, 2.860558832897748, 3.059036517563198, 3.933998167257511, 4.646074116970887, 5.604132935161850, 5.985344757508338, 6.859109284234373, 7.214247533262595, 7.823261341310853, 8.893422377651092, 9.280172451423393, 9.788693095017872, 10.22417099761109, 10.96739546134274, 11.85605423359099, 12.31730339438544, 12.45717177400896, 13.35715336481839, 13.93940343252848, 14.41677319956909, 14.92405090866785, 15.70722104986870, 16.10839354723984
