| L(s) = 1 | + 2-s + 3·3-s + 4-s − 5-s + 3·6-s + 8-s + 6·9-s − 10-s − 2·11-s + 3·12-s − 3·15-s + 16-s − 4·17-s + 6·18-s − 6·19-s − 20-s − 2·22-s + 3·23-s + 3·24-s + 25-s + 9·27-s + 9·29-s − 3·30-s − 4·31-s + 32-s − 6·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s + 0.866·12-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 1.41·18-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 0.625·23-s + 0.612·24-s + 1/5·25-s + 1.73·27-s + 1.67·29-s − 0.547·30-s − 0.718·31-s + 0.176·32-s − 1.04·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.283859656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.283859656\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.86020881619906947342327663044, −10.12095915307289712181643984209, −8.801329180936076378169644135886, −8.417995701997192590477084041039, −7.36597660738767186076247894677, −6.57205733073059424709937866086, −4.88294429143581850116350412354, −4.00544203434240366935428278549, −3.00212844655097191031458367831, −2.07618093352752431919693916301,
2.07618093352752431919693916301, 3.00212844655097191031458367831, 4.00544203434240366935428278549, 4.88294429143581850116350412354, 6.57205733073059424709937866086, 7.36597660738767186076247894677, 8.417995701997192590477084041039, 8.801329180936076378169644135886, 10.12095915307289712181643984209, 10.86020881619906947342327663044
