| L(s) = 1 | + 5.02·2-s + 3·3-s + 17.2·4-s − 20.4·5-s + 15.0·6-s + 46.2·8-s + 9·9-s − 102.·10-s − 11·11-s + 51.6·12-s + 0.0115·13-s − 61.3·15-s + 94.7·16-s + 9.52·17-s + 45.1·18-s + 93.4·19-s − 351.·20-s − 55.2·22-s + 99.9·23-s + 138.·24-s + 292.·25-s + 0.0580·26-s + 27·27-s + 276.·29-s − 307.·30-s + 181.·31-s + 105.·32-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 0.577·3-s + 2.15·4-s − 1.82·5-s + 1.02·6-s + 2.04·8-s + 0.333·9-s − 3.24·10-s − 0.301·11-s + 1.24·12-s + 0.000246·13-s − 1.05·15-s + 1.47·16-s + 0.135·17-s + 0.591·18-s + 1.12·19-s − 3.93·20-s − 0.535·22-s + 0.906·23-s + 1.18·24-s + 2.34·25-s + 0.000437·26-s + 0.192·27-s + 1.76·29-s − 1.87·30-s + 1.05·31-s + 0.581·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.464247896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.464247896\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.02T + 8T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 13 | \( 1 - 0.0115T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.52T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 404.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 27.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 76.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 585.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 530.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 354.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 471.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 563.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 895.T + 9.12e5T^{2} \) |
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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
−8.665326838980648939357153764896, −8.056034686280299192137238460947, −7.13066022411030263051852686375, −6.80032933903810685620477360103, −5.36420504592154321386504198399, −4.71872528433175430424712910841, −3.94099317815684095627605309224, −3.26252258747454610775213609098, −2.64305606215935869645640894489, −0.928216583144045053278220163067,
0.928216583144045053278220163067, 2.64305606215935869645640894489, 3.26252258747454610775213609098, 3.94099317815684095627605309224, 4.71872528433175430424712910841, 5.36420504592154321386504198399, 6.80032933903810685620477360103, 7.13066022411030263051852686375, 8.056034686280299192137238460947, 8.665326838980648939357153764896
