| L(s) = 1 | + 2·2-s − 3-s + 2·4-s + 5-s − 2·6-s + 9-s + 2·10-s + 11-s − 2·12-s + 2·13-s − 15-s − 4·16-s − 2·17-s + 2·18-s − 6·19-s + 2·20-s + 2·22-s + 3·23-s − 4·25-s + 4·26-s − 27-s + 3·29-s − 2·30-s − 2·31-s − 8·32-s − 33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 0.258·15-s − 16-s − 0.485·17-s + 0.471·18-s − 1.37·19-s + 0.447·20-s + 0.426·22-s + 0.625·23-s − 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.557·29-s − 0.365·30-s − 0.359·31-s − 1.41·32-s − 0.174·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6909 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6909 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.15643207980329887502214707718, −6.46427393643572557800153596855, −6.26601972151855913723573754005, −5.26847892357043316575151303157, −4.93665543394919174915106478313, −3.99784950655882501004286576008, −3.51543089827414755770668868445, −2.40030582023801117847243436493, −1.60587301327612646198388401084, 0,
1.60587301327612646198388401084, 2.40030582023801117847243436493, 3.51543089827414755770668868445, 3.99784950655882501004286576008, 4.93665543394919174915106478313, 5.26847892357043316575151303157, 6.26601972151855913723573754005, 6.46427393643572557800153596855, 7.15643207980329887502214707718
