| L(s) = 1 | − 1.05·3-s − 1.02·5-s − 1.89·9-s − 1.28·11-s − 2.29·13-s + 1.08·15-s + 4.44·17-s − 2.95·19-s + 4.81·23-s − 3.94·25-s + 5.14·27-s + 0.100·29-s + 0.925·31-s + 1.35·33-s − 7.07·37-s + 2.41·39-s + 41-s − 5.47·43-s + 1.95·45-s − 2.99·47-s − 4.67·51-s − 1.46·53-s + 1.32·55-s + 3.10·57-s + 4.37·59-s − 5.39·61-s + 2.36·65-s + ⋯ |
| L(s) = 1 | − 0.606·3-s − 0.460·5-s − 0.631·9-s − 0.387·11-s − 0.637·13-s + 0.279·15-s + 1.07·17-s − 0.676·19-s + 1.00·23-s − 0.788·25-s + 0.990·27-s + 0.0187·29-s + 0.166·31-s + 0.235·33-s − 1.16·37-s + 0.386·39-s + 0.156·41-s − 0.834·43-s + 0.290·45-s − 0.437·47-s − 0.654·51-s − 0.200·53-s + 0.178·55-s + 0.410·57-s + 0.569·59-s − 0.690·61-s + 0.293·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7215862536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7215862536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 1.05T + 3T^{2} \) |
| 5 | \( 1 + 1.02T + 5T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 2.95T + 19T^{2} \) |
| 23 | \( 1 - 4.81T + 23T^{2} \) |
| 29 | \( 1 - 0.100T + 29T^{2} \) |
| 31 | \( 1 - 0.925T + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 + 2.99T + 47T^{2} \) |
| 53 | \( 1 + 1.46T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 5.18T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 9.89T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 - 3.20T + 97T^{2} \) |
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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
−7.72444242401781594088660886642, −7.21989528099057869034138085868, −6.37511799108568699422127776154, −5.70288620582030524216699761618, −5.08989879092242047781680507372, −4.45750705080062749943902546268, −3.40421919666076116487251496734, −2.82723245365249182891365418333, −1.68725511538766507601537673522, −0.42589246309817497616644209910,
0.42589246309817497616644209910, 1.68725511538766507601537673522, 2.82723245365249182891365418333, 3.40421919666076116487251496734, 4.45750705080062749943902546268, 5.08989879092242047781680507372, 5.70288620582030524216699761618, 6.37511799108568699422127776154, 7.21989528099057869034138085868, 7.72444242401781594088660886642
