| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s − 2·9-s − 11-s − 12-s − 2·13-s + 14-s + 16-s − 7·17-s − 2·18-s − 4·19-s − 21-s − 22-s + 8·23-s − 24-s − 2·26-s + 5·27-s + 28-s + 6·29-s + 5·31-s + 32-s + 33-s − 7·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.69·17-s − 0.471·18-s − 0.917·19-s − 0.218·21-s − 0.213·22-s + 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.962·27-s + 0.188·28-s + 1.11·29-s + 0.898·31-s + 0.176·32-s + 0.174·33-s − 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.076383951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076383951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 13 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−8.613629416979675724804802254989, −7.56692795459538108836532288760, −6.77387969307226735079997824734, −6.25529707028001742635125379992, −5.38042874658420008327088341294, −4.73243614418658098276908600500, −4.18754903046365038975951219332, −2.79405002421665786398032483375, −2.34037668650565508224723806404, −0.75003717916789428506503125454,
0.75003717916789428506503125454, 2.34037668650565508224723806404, 2.79405002421665786398032483375, 4.18754903046365038975951219332, 4.73243614418658098276908600500, 5.38042874658420008327088341294, 6.25529707028001742635125379992, 6.77387969307226735079997824734, 7.56692795459538108836532288760, 8.613629416979675724804802254989
