| L(s) = 1 | + 1.23·3-s + 1.23·5-s − 7-s − 1.47·9-s − 4·11-s − 1.23·13-s + 1.52·15-s − 2·17-s − 1.23·19-s − 1.23·21-s − 6.47·23-s − 3.47·25-s − 5.52·27-s + 1.52·29-s − 4.94·33-s − 1.23·35-s + 6.47·37-s − 1.52·39-s − 2·41-s − 8.94·43-s − 1.81·45-s + 12.9·47-s + 49-s − 2.47·51-s − 8.94·53-s − 4.94·55-s − 1.52·57-s + ⋯ |
| L(s) = 1 | + 0.713·3-s + 0.552·5-s − 0.377·7-s − 0.490·9-s − 1.20·11-s − 0.342·13-s + 0.394·15-s − 0.485·17-s − 0.283·19-s − 0.269·21-s − 1.34·23-s − 0.694·25-s − 1.06·27-s + 0.283·29-s − 0.860·33-s − 0.208·35-s + 1.06·37-s − 0.244·39-s − 0.312·41-s − 1.36·43-s − 0.271·45-s + 1.88·47-s + 0.142·49-s − 0.346·51-s − 1.22·53-s − 0.666·55-s − 0.202·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.881014942611411619384651300057, −8.086577700238067299655157450306, −7.56633626263904897671942658996, −6.35250092801980494867377025217, −5.75461554987906457195854636981, −4.78506457139497289772517680456, −3.67256178251561749008942932086, −2.65590677722670881435561590895, −2.06135426230661848811073629963, 0,
2.06135426230661848811073629963, 2.65590677722670881435561590895, 3.67256178251561749008942932086, 4.78506457139497289772517680456, 5.75461554987906457195854636981, 6.35250092801980494867377025217, 7.56633626263904897671942658996, 8.086577700238067299655157450306, 8.881014942611411619384651300057
