| 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
| \(L(1)\) |
\(\approx\) |
\(1.409332007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409332007\) |
| \(L(\frac{3}{2})\) |
|
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
−11.06289048806052475370595948438, −10.36383578936153834327307787127, −9.635128409981478362582371513154, −7.916325009885536263558334349206, −6.89394037901040901071472214534, −6.09157415694094013326272320913, −5.26789391708575690709750011159, −4.72473630829615747045956251199, −3.08854077694740486482752774267, −1.16479363083596500346798967246,
1.16479363083596500346798967246, 3.08854077694740486482752774267, 4.72473630829615747045956251199, 5.26789391708575690709750011159, 6.09157415694094013326272320913, 6.89394037901040901071472214534, 7.916325009885536263558334349206, 9.635128409981478362582371513154, 10.36383578936153834327307787127, 11.06289048806052475370595948438
