| L(s) = 1 | + 3-s − 5-s + 9-s + 2·11-s + 4·13-s − 15-s − 2·17-s + 2·23-s + 25-s + 27-s + 6·29-s − 4·31-s + 2·33-s − 10·37-s + 4·39-s + 2·41-s + 4·43-s − 45-s + 8·47-s − 2·51-s + 10·53-s − 2·55-s − 4·59-s − 8·61-s − 4·65-s + 8·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.348·33-s − 1.64·37-s + 0.640·39-s + 0.312·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 1.37·53-s − 0.269·55-s − 0.520·59-s − 1.02·61-s − 0.496·65-s + 0.977·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.316138848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316138848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.893489777978544286027993134454, −8.108904328695199939238514906284, −7.24955562027682389525162465562, −6.62715181950280608104035419145, −5.75581707431863410403599664373, −4.68632220114176897935789216298, −3.88565765891595445753004910489, −3.25117996615682924834280622800, −2.09932124184198411384028640861, −0.947145747322200559624682748248,
0.947145747322200559624682748248, 2.09932124184198411384028640861, 3.25117996615682924834280622800, 3.88565765891595445753004910489, 4.68632220114176897935789216298, 5.75581707431863410403599664373, 6.62715181950280608104035419145, 7.24955562027682389525162465562, 8.108904328695199939238514906284, 8.893489777978544286027993134454
