| L(s) = 1 | − 3-s − 2·4-s + 2·7-s + 9-s + 2·11-s + 2·12-s + 6·13-s + 4·16-s − 2·17-s + 5·19-s − 2·21-s + 5·23-s − 27-s − 4·28-s + 2·29-s − 2·33-s − 2·36-s − 7·37-s − 6·39-s + 12·41-s + 4·43-s − 4·44-s + 47-s − 4·48-s − 3·49-s + 2·51-s − 12·52-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s + 1.66·13-s + 16-s − 0.485·17-s + 1.14·19-s − 0.436·21-s + 1.04·23-s − 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.348·33-s − 1/3·36-s − 1.15·37-s − 0.960·39-s + 1.87·41-s + 0.609·43-s − 0.603·44-s + 0.145·47-s − 0.577·48-s − 3/7·49-s + 0.280·51-s − 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.717640292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.717640292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 67 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−8.403254904850555275319386961195, −7.54139076365234356889739709355, −6.78955229343358044525119304796, −5.82478890277641366069790362316, −5.41044393224517701467605503454, −4.47708486185994989422607474153, −3.99482264029090563878758260294, −3.05491407019668575718106042972, −1.46080622704783683345621895248, −0.860191810638515821268156082604,
0.860191810638515821268156082604, 1.46080622704783683345621895248, 3.05491407019668575718106042972, 3.99482264029090563878758260294, 4.47708486185994989422607474153, 5.41044393224517701467605503454, 5.82478890277641366069790362316, 6.78955229343358044525119304796, 7.54139076365234356889739709355, 8.403254904850555275319386961195
