| L(s) = 1 | + 3·11-s + 2·17-s + 2·19-s + 6·23-s − 5·25-s + 2·29-s + 4·31-s + 37-s + 7·41-s + 4·43-s + 47-s − 9·53-s + 8·59-s + 4·61-s + 12·67-s + 5·71-s + 13·73-s − 10·79-s − 83-s − 2·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 0.485·17-s + 0.458·19-s + 1.25·23-s − 25-s + 0.371·29-s + 0.718·31-s + 0.164·37-s + 1.09·41-s + 0.609·43-s + 0.145·47-s − 1.23·53-s + 1.04·59-s + 0.512·61-s + 1.46·67-s + 0.593·71-s + 1.52·73-s − 1.12·79-s − 0.109·83-s − 0.211·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.582064469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.582064469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.51246782976270, −12.96925051284112, −12.51764498913636, −11.99369587428964, −11.61974728455307, −11.03967701244580, −10.75675782996872, −9.905929569338328, −9.582347356715323, −9.268835789646658, −8.546965139110718, −8.105905822148848, −7.605947567595331, −6.966253509327213, −6.609547823426956, −5.997182862283542, −5.455715028448290, −4.952692017295360, −4.241364376708541, −3.836092080919163, −3.170275929117668, −2.611180411356872, −1.889904953303220, −1.092522823514302, −0.6658066783466515,
0.6658066783466515, 1.092522823514302, 1.889904953303220, 2.611180411356872, 3.170275929117668, 3.836092080919163, 4.241364376708541, 4.952692017295360, 5.455715028448290, 5.997182862283542, 6.609547823426956, 6.966253509327213, 7.605947567595331, 8.105905822148848, 8.546965139110718, 9.268835789646658, 9.582347356715323, 9.905929569338328, 10.75675782996872, 11.03967701244580, 11.61974728455307, 11.99369587428964, 12.51764498913636, 12.96925051284112, 13.51246782976270
