| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s + 5·13-s + 19-s − 2·21-s − 3·23-s + 27-s + 3·29-s − 31-s − 33-s + 2·37-s + 5·39-s − 6·41-s + 11·43-s + 12·47-s − 3·49-s + 12·53-s + 57-s − 6·59-s − 2·61-s − 2·63-s + 2·67-s − 3·69-s − 9·71-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.229·19-s − 0.436·21-s − 0.625·23-s + 0.192·27-s + 0.557·29-s − 0.179·31-s − 0.174·33-s + 0.328·37-s + 0.800·39-s − 0.937·41-s + 1.67·43-s + 1.75·47-s − 3/7·49-s + 1.64·53-s + 0.132·57-s − 0.781·59-s − 0.256·61-s − 0.251·63-s + 0.244·67-s − 0.361·69-s − 1.06·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.915524743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.915524743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.34162091064488, −13.80210936756460, −13.58762364746105, −12.98268339215537, −12.54286812376516, −11.95466762589130, −11.42731035252200, −10.61488311392620, −10.44601640596286, −9.790621250622705, −9.098504201298048, −8.861156762216983, −8.229746252493405, −7.627544888169612, −7.174343466991579, −6.373569975063660, −6.051874986711703, −5.434039968677406, −4.595244614412349, −3.909548552063145, −3.547963355299746, −2.811746172268940, −2.256073107278249, −1.344998749869644, −0.6017992019919155,
0.6017992019919155, 1.344998749869644, 2.256073107278249, 2.811746172268940, 3.547963355299746, 3.909548552063145, 4.595244614412349, 5.434039968677406, 6.051874986711703, 6.373569975063660, 7.174343466991579, 7.627544888169612, 8.229746252493405, 8.861156762216983, 9.098504201298048, 9.790621250622705, 10.44601640596286, 10.61488311392620, 11.42731035252200, 11.95466762589130, 12.54286812376516, 12.98268339215537, 13.58762364746105, 13.80210936756460, 14.34162091064488
