| L(s) = 1 | + 2-s + 3-s + 4-s − 1.41·5-s + 6-s + 8-s + 9-s − 1.41·10-s + 0.414·11-s + 12-s + 13-s − 1.41·15-s + 16-s + 5.24·17-s + 18-s + 3·19-s − 1.41·20-s + 0.414·22-s − 9.07·23-s + 24-s − 2.99·25-s + 26-s + 27-s + 5·29-s − 1.41·30-s + 6.82·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.632·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.447·10-s + 0.124·11-s + 0.288·12-s + 0.277·13-s − 0.365·15-s + 0.250·16-s + 1.27·17-s + 0.235·18-s + 0.688·19-s − 0.316·20-s + 0.0883·22-s − 1.89·23-s + 0.204·24-s − 0.599·25-s + 0.196·26-s + 0.192·27-s + 0.928·29-s − 0.258·30-s + 1.22·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.618211501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.618211501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 6.24T + 89T^{2} \) |
| 97 | \( 1 - 7.89T + 97T^{2} \) |
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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
−8.255778077666497004749933639107, −7.75434901552065513055240273509, −7.15945700935716349767501988009, −6.10927765434231720498422430719, −5.56691224608062999877867274729, −4.43417419444974324714972749693, −3.89130447155569416007337386990, −3.15891926873810763346200946546, −2.24084956120762708570795613312, −1.00598125479172727285268592316,
1.00598125479172727285268592316, 2.24084956120762708570795613312, 3.15891926873810763346200946546, 3.89130447155569416007337386990, 4.43417419444974324714972749693, 5.56691224608062999877867274729, 6.10927765434231720498422430719, 7.15945700935716349767501988009, 7.75434901552065513055240273509, 8.255778077666497004749933639107
