| L(s) = 1 | − 3-s + 2·5-s + 2·7-s + 9-s + 2·13-s − 2·15-s − 6·17-s + 2·19-s − 2·21-s − 25-s − 27-s − 6·29-s + 4·35-s + 4·37-s − 2·39-s + 2·41-s − 6·43-s + 2·45-s − 4·47-s − 3·49-s + 6·51-s + 6·53-s − 2·57-s − 4·61-s + 2·63-s + 4·65-s − 10·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s + 0.657·37-s − 0.320·39-s + 0.312·41-s − 0.914·43-s + 0.298·45-s − 0.583·47-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.264·57-s − 0.512·61-s + 0.251·63-s + 0.496·65-s − 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.63924216711514, −16.08783139007999, −15.44202316419976, −14.91720327818573, −14.29572298657197, −13.62096496629308, −13.22275174810901, −12.79178109165985, −11.77785110525820, −11.42951022119086, −10.94717980179883, −10.29344811308604, −9.678699640468210, −9.070112578400939, −8.491004722925879, −7.717913730550358, −7.052165889605339, −6.330280413130660, −5.834880412856743, −5.205390335903918, −4.533501391432350, −3.869331822728176, −2.771459661327477, −1.888524572233383, −1.355027243707293, 0,
1.355027243707293, 1.888524572233383, 2.771459661327477, 3.869331822728176, 4.533501391432350, 5.205390335903918, 5.834880412856743, 6.330280413130660, 7.052165889605339, 7.717913730550358, 8.491004722925879, 9.070112578400939, 9.678699640468210, 10.29344811308604, 10.94717980179883, 11.42951022119086, 11.77785110525820, 12.79178109165985, 13.22275174810901, 13.62096496629308, 14.29572298657197, 14.91720327818573, 15.44202316419976, 16.08783139007999, 16.63924216711514
