| L(s) = 1 | − 2.32·2-s − 3·3-s − 2.57·4-s − 5·5-s + 6.98·6-s + 24.6·8-s + 9·9-s + 11.6·10-s + 52.6·11-s + 7.73·12-s − 78.1·13-s + 15·15-s − 36.7·16-s + 76.1·17-s − 20.9·18-s − 73.8·19-s + 12.8·20-s − 122.·22-s + 137.·23-s − 73.8·24-s + 25·25-s + 182.·26-s − 27·27-s − 289.·29-s − 34.9·30-s + 83.9·31-s − 111.·32-s + ⋯ |
| L(s) = 1 | − 0.823·2-s − 0.577·3-s − 0.322·4-s − 0.447·5-s + 0.475·6-s + 1.08·8-s + 0.333·9-s + 0.368·10-s + 1.44·11-s + 0.186·12-s − 1.66·13-s + 0.258·15-s − 0.574·16-s + 1.08·17-s − 0.274·18-s − 0.892·19-s + 0.144·20-s − 1.18·22-s + 1.25·23-s − 0.628·24-s + 0.200·25-s + 1.37·26-s − 0.192·27-s − 1.85·29-s − 0.212·30-s + 0.486·31-s − 0.615·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6142821607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6142821607\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.32T + 8T^{2} \) |
| 11 | \( 1 - 52.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 78.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 289.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 324.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 392.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 459.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 494.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 88.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 641.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 672.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 456.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 604.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 868.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 126.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 289.T + 9.12e5T^{2} \) |
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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
−9.791629285685427396301272924939, −9.299025697396090363861341555212, −8.356499894665962983358668168581, −7.36198856867049039050792492127, −6.82363541675593018287473658891, −5.39574348618124850871360967461, −4.57863605538812904309095350011, −3.57689704710759379144958574584, −1.74148627855439960545428395030, −0.53712849838396446595494277940,
0.53712849838396446595494277940, 1.74148627855439960545428395030, 3.57689704710759379144958574584, 4.57863605538812904309095350011, 5.39574348618124850871360967461, 6.82363541675593018287473658891, 7.36198856867049039050792492127, 8.356499894665962983358668168581, 9.299025697396090363861341555212, 9.791629285685427396301272924939
