| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s − 8-s − 2·9-s − 2·10-s + 2·11-s + 12-s − 2·13-s + 2·15-s + 16-s + 2·18-s + 3·19-s + 2·20-s − 2·22-s − 24-s − 25-s + 2·26-s − 5·27-s + 3·29-s − 2·30-s + 5·31-s − 32-s + 2·33-s − 2·36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.603·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 0.471·18-s + 0.688·19-s + 0.447·20-s − 0.426·22-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.962·27-s + 0.557·29-s − 0.365·30-s + 0.898·31-s − 0.176·32-s + 0.348·33-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.34540393211344, −15.03095685496364, −14.27680076505199, −13.94586430774779, −13.62972966649142, −12.75293338276278, −12.26279515209896, −11.53375637290738, −11.28668238711226, −10.39570250601516, −9.883166118395855, −9.497069280344294, −9.039007931052646, −8.435034194552511, −7.872901901268276, −7.377140483083622, −6.447702631988260, −6.232606962616502, −5.428546377491820, −4.820016532739918, −3.894108841072304, −3.038120909013673, −2.637612764918161, −1.838235022268020, −1.174417445383328, 0,
1.174417445383328, 1.838235022268020, 2.637612764918161, 3.038120909013673, 3.894108841072304, 4.820016532739918, 5.428546377491820, 6.232606962616502, 6.447702631988260, 7.377140483083622, 7.872901901268276, 8.435034194552511, 9.039007931052646, 9.497069280344294, 9.883166118395855, 10.39570250601516, 11.28668238711226, 11.53375637290738, 12.26279515209896, 12.75293338276278, 13.62972966649142, 13.94586430774779, 14.27680076505199, 15.03095685496364, 15.34540393211344
