| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 4·11-s − 2·13-s − 15-s − 6·17-s + 4·19-s − 21-s + 8·23-s + 25-s − 27-s − 2·29-s − 4·33-s + 35-s − 2·37-s + 2·39-s + 10·41-s + 4·43-s + 45-s + 49-s + 6·51-s + 14·53-s + 4·55-s − 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 0.840·51-s + 1.92·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527981036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527981036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−10.24217028435075132499557008187, −9.260421204459754863440873199679, −8.814898165377068035765137479106, −7.34135464290359642603613929036, −6.81120762098947234121801809812, −5.78942030989278693794781516494, −4.91268225593800426125608367296, −3.99697191426783157831064172663, −2.48023092517801384595443924306, −1.11320082817149406856784190979,
1.11320082817149406856784190979, 2.48023092517801384595443924306, 3.99697191426783157831064172663, 4.91268225593800426125608367296, 5.78942030989278693794781516494, 6.81120762098947234121801809812, 7.34135464290359642603613929036, 8.814898165377068035765137479106, 9.260421204459754863440873199679, 10.24217028435075132499557008187
