| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 4·11-s + 5·13-s + 15-s − 2·17-s − 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s + 3·29-s + 6·31-s + 4·33-s − 2·35-s − 37-s − 5·39-s + 11·43-s − 45-s − 9·47-s − 3·49-s + 2·51-s + 3·53-s + 4·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.38·13-s + 0.258·15-s − 0.485·17-s − 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.07·31-s + 0.696·33-s − 0.338·35-s − 0.164·37-s − 0.800·39-s + 1.67·43-s − 0.149·45-s − 1.31·47-s − 3/7·49-s + 0.280·51-s + 0.412·53-s + 0.539·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.222293412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222293412\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.916677972317757056748609849746, −7.05090806336895782060969168250, −6.18898407507515532185632054927, −5.87072983216039968743262262563, −4.79605048996490369391618777683, −4.43454903527971835312277545794, −3.61889172978127532334328001434, −2.53450318587269765998204835036, −1.70238625993063668905497055006, −0.55198894690984978554393837133,
0.55198894690984978554393837133, 1.70238625993063668905497055006, 2.53450318587269765998204835036, 3.61889172978127532334328001434, 4.43454903527971835312277545794, 4.79605048996490369391618777683, 5.87072983216039968743262262563, 6.18898407507515532185632054927, 7.05090806336895782060969168250, 7.916677972317757056748609849746
