| L(s) = 1 | + 3·3-s − 5-s + 7-s + 6·9-s + 4·11-s − 13-s − 3·15-s − 7·17-s − 19-s + 3·21-s + 5·23-s + 25-s + 9·27-s − 7·29-s + 2·31-s + 12·33-s − 35-s + 6·37-s − 3·39-s + 6·41-s + 10·43-s − 6·45-s + 8·47-s − 6·49-s − 21·51-s + 3·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.20·11-s − 0.277·13-s − 0.774·15-s − 1.69·17-s − 0.229·19-s + 0.654·21-s + 1.04·23-s + 1/5·25-s + 1.73·27-s − 1.29·29-s + 0.359·31-s + 2.08·33-s − 0.169·35-s + 0.986·37-s − 0.480·39-s + 0.937·41-s + 1.52·43-s − 0.894·45-s + 1.16·47-s − 6/7·49-s − 2.94·51-s + 0.412·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.094671036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.094671036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.267219893872805567185632491569, −7.34200556285541191824996664454, −7.04833935174515953449271499984, −6.12955941478866655435988776763, −4.86266215491536205379785389384, −4.07697367631552494521103646729, −3.80601356115634372872218732674, −2.63289917628274243452814326466, −2.15275573824262558078973699138, −1.02026705389525234754030645886,
1.02026705389525234754030645886, 2.15275573824262558078973699138, 2.63289917628274243452814326466, 3.80601356115634372872218732674, 4.07697367631552494521103646729, 4.86266215491536205379785389384, 6.12955941478866655435988776763, 7.04833935174515953449271499984, 7.34200556285541191824996664454, 8.267219893872805567185632491569
