| L(s) = 1 | + 5-s + 7-s + 2·11-s − 4·13-s − 3·17-s − 2·19-s − 6·23-s − 4·25-s − 6·29-s + 4·31-s + 35-s − 11·37-s − 5·41-s + 7·43-s − 5·47-s + 49-s + 6·53-s + 2·55-s − 9·59-s + 6·61-s − 4·65-s + 4·67-s − 8·71-s − 2·73-s + 2·77-s + 11·79-s + 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.603·11-s − 1.10·13-s − 0.727·17-s − 0.458·19-s − 1.25·23-s − 4/5·25-s − 1.11·29-s + 0.718·31-s + 0.169·35-s − 1.80·37-s − 0.780·41-s + 1.06·43-s − 0.729·47-s + 1/7·49-s + 0.824·53-s + 0.269·55-s − 1.17·59-s + 0.768·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s − 0.234·73-s + 0.227·77-s + 1.23·79-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.384100142784634694883412443428, −7.58302186141675429871193910609, −6.83745891916597366541417972323, −6.07362576682198052110145132919, −5.27831389557032756698974946087, −4.43938661360568317546127663262, −3.65950256859576512702489926953, −2.34189561169258657657436515538, −1.72935093915045004554881157865, 0,
1.72935093915045004554881157865, 2.34189561169258657657436515538, 3.65950256859576512702489926953, 4.43938661360568317546127663262, 5.27831389557032756698974946087, 6.07362576682198052110145132919, 6.83745891916597366541417972323, 7.58302186141675429871193910609, 8.384100142784634694883412443428
