| L(s) = 1 | − 0.541·2-s − 7.70·4-s − 16.7·5-s + 8.50·8-s + 9.06·10-s + 26.1·11-s − 34.4·13-s + 57.0·16-s − 81.5·17-s − 96.8·19-s + 128.·20-s − 14.1·22-s − 183.·23-s + 154.·25-s + 18.6·26-s + 42.2·29-s − 279.·31-s − 98.9·32-s + 44.1·34-s − 48.6·37-s + 52.4·38-s − 142.·40-s − 4.73·41-s − 419.·43-s − 201.·44-s + 99.6·46-s + 39.9·47-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 0.963·4-s − 1.49·5-s + 0.375·8-s + 0.286·10-s + 0.717·11-s − 0.735·13-s + 0.891·16-s − 1.16·17-s − 1.16·19-s + 1.44·20-s − 0.137·22-s − 1.66·23-s + 1.23·25-s + 0.140·26-s + 0.270·29-s − 1.62·31-s − 0.546·32-s + 0.222·34-s − 0.216·37-s + 0.224·38-s − 0.562·40-s − 0.0180·41-s − 1.48·43-s − 0.691·44-s + 0.319·46-s + 0.124·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08005750049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08005750049\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.541T + 8T^{2} \) |
| 5 | \( 1 + 16.7T + 125T^{2} \) |
| 11 | \( 1 - 26.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 48.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.73T + 6.89e4T^{2} \) |
| 43 | \( 1 + 419.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 39.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 287.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 465.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 242.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 634.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 403.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 308.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 798.T + 9.12e5T^{2} \) |
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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
−8.987823904737482300927843948298, −8.566059363387391720638966851095, −7.74526925708738626720014488398, −7.03796207928730025474725027371, −5.94948657026108229601006804560, −4.60273908838488897353345818808, −4.24948496420171035446050571727, −3.42256628546404641125910549940, −1.83424885731413421072743911659, −0.14331402595706883418542386070,
0.14331402595706883418542386070, 1.83424885731413421072743911659, 3.42256628546404641125910549940, 4.24948496420171035446050571727, 4.60273908838488897353345818808, 5.94948657026108229601006804560, 7.03796207928730025474725027371, 7.74526925708738626720014488398, 8.566059363387391720638966851095, 8.987823904737482300927843948298
