| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·13-s − 15-s + 6·19-s + 2·21-s + 8·23-s + 25-s − 27-s + 8·29-s + 4·31-s − 2·35-s − 6·37-s − 2·39-s + 12·41-s − 6·43-s + 45-s − 3·49-s + 2·53-s − 6·57-s − 4·59-s − 2·61-s − 2·63-s + 2·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.37·19-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.338·35-s − 0.986·37-s − 0.320·39-s + 1.87·41-s − 0.914·43-s + 0.149·45-s − 3/7·49-s + 0.274·53-s − 0.794·57-s − 0.520·59-s − 0.256·61-s − 0.251·63-s + 0.248·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.69367590300738, −13.83683267126190, −13.61732765104814, −13.11892627776778, −12.58963573405725, −11.98600644982192, −11.70976203307348, −10.84910126780147, −10.65799394473651, −9.946033659996525, −9.574150283741040, −8.968731893212113, −8.553707322077603, −7.669081671561467, −7.219391241163428, −6.543058372130765, −6.272974640316468, −5.571268263235958, −5.065574989100860, −4.523079555506621, −3.723838710125862, −2.911067092124756, −2.763801754268101, −1.340403661661271, −1.121924228634820, 0,
1.121924228634820, 1.340403661661271, 2.763801754268101, 2.911067092124756, 3.723838710125862, 4.523079555506621, 5.065574989100860, 5.571268263235958, 6.272974640316468, 6.543058372130765, 7.219391241163428, 7.669081671561467, 8.553707322077603, 8.968731893212113, 9.574150283741040, 9.946033659996525, 10.65799394473651, 10.84910126780147, 11.70976203307348, 11.98600644982192, 12.58963573405725, 13.11892627776778, 13.61732765104814, 13.83683267126190, 14.69367590300738
