| L(s) = 1 | + 0.915·2-s + 4.67·3-s − 7.16·4-s + 20.1·5-s + 4.28·6-s + 19.4·7-s − 13.8·8-s − 5.10·9-s + 18.4·10-s − 17.9·11-s − 33.5·12-s + 35.3·13-s + 17.7·14-s + 94.1·15-s + 44.5·16-s + 93.9·17-s − 4.67·18-s − 67.7·19-s − 144.·20-s + 90.9·21-s − 16.3·22-s + 23·23-s − 64.9·24-s + 279.·25-s + 32.3·26-s − 150.·27-s − 139.·28-s + ⋯ |
| L(s) = 1 | + 0.323·2-s + 0.900·3-s − 0.895·4-s + 1.79·5-s + 0.291·6-s + 1.04·7-s − 0.613·8-s − 0.189·9-s + 0.582·10-s − 0.490·11-s − 0.806·12-s + 0.753·13-s + 0.339·14-s + 1.62·15-s + 0.696·16-s + 1.34·17-s − 0.0611·18-s − 0.818·19-s − 1.61·20-s + 0.945·21-s − 0.158·22-s + 0.208·23-s − 0.552·24-s + 2.23·25-s + 0.243·26-s − 1.07·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.119765300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.119765300\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 0.915T + 8T^{2} \) |
| 3 | \( 1 - 4.67T + 27T^{2} \) |
| 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 - 19.4T + 343T^{2} \) |
| 11 | \( 1 + 17.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.7T + 6.85e3T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 188.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 110.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 248.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 191.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 72.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 969.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 158.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 156.T + 9.12e5T^{2} \) |
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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
−10.00244970605583442767085099956, −9.130901338415931363509146156343, −8.544282621628014442971201697574, −7.84354525283357494601068250015, −6.19711342312226193984554197417, −5.51772239498739646663655029276, −4.72867200677310392068512020790, −3.36354263671845917589219568835, −2.33155426497461785464317337628, −1.21695931659486889451442800377,
1.21695931659486889451442800377, 2.33155426497461785464317337628, 3.36354263671845917589219568835, 4.72867200677310392068512020790, 5.51772239498739646663655029276, 6.19711342312226193984554197417, 7.84354525283357494601068250015, 8.544282621628014442971201697574, 9.130901338415931363509146156343, 10.00244970605583442767085099956
