| L(s) = 1 | + 3-s + 3.35·5-s − 2.24·7-s + 9-s − 4.93·11-s + 3.35·15-s + 0.911·17-s + 3.80·19-s − 2.24·21-s − 2.02·23-s + 6.26·25-s + 27-s − 3.93·29-s + 8.82·31-s − 4.93·33-s − 7.54·35-s + 8.80·37-s + 6.93·41-s + 2.28·43-s + 3.35·45-s − 3.80·47-s − 1.95·49-s + 0.911·51-s + 0.542·53-s − 16.5·55-s + 3.80·57-s + 4.71·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.50·5-s − 0.849·7-s + 0.333·9-s − 1.48·11-s + 0.866·15-s + 0.221·17-s + 0.872·19-s − 0.490·21-s − 0.422·23-s + 1.25·25-s + 0.192·27-s − 0.731·29-s + 1.58·31-s − 0.859·33-s − 1.27·35-s + 1.44·37-s + 1.08·41-s + 0.348·43-s + 0.500·45-s − 0.554·47-s − 0.278·49-s + 0.127·51-s + 0.0745·53-s − 2.23·55-s + 0.503·57-s + 0.613·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.989808856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.989808856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 4.93T + 11T^{2} \) |
| 17 | \( 1 - 0.911T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 + 3.80T + 47T^{2} \) |
| 53 | \( 1 - 0.542T + 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 3.67T + 61T^{2} \) |
| 67 | \( 1 - 1.52T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 - 3.74T + 79T^{2} \) |
| 83 | \( 1 + 2.30T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
| 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
−7.87231033552075609638572441657, −7.16148513087651379598916558046, −6.31378264526117820596925452133, −5.79034180529840528667960542240, −5.19918322558691087130442196889, −4.30370472798702039245228529742, −3.15926911772048801436923223337, −2.67972916556820626515991478310, −2.01459637560003668229707804025, −0.818535051315247205296316266835,
0.818535051315247205296316266835, 2.01459637560003668229707804025, 2.67972916556820626515991478310, 3.15926911772048801436923223337, 4.30370472798702039245228529742, 5.19918322558691087130442196889, 5.79034180529840528667960542240, 6.31378264526117820596925452133, 7.16148513087651379598916558046, 7.87231033552075609638572441657
