| L(s) = 1 | + 2.69·2-s + 5.19·3-s − 0.731·4-s − 5.54·5-s + 14.0·6-s − 23.5·8-s + 0.0251·9-s − 14.9·10-s + 25.5·11-s − 3.80·12-s − 56.4·13-s − 28.8·15-s − 57.6·16-s − 22.0·17-s + 0.0677·18-s − 1.02·19-s + 4.05·20-s + 68.9·22-s + 121.·23-s − 122.·24-s − 94.2·25-s − 152.·26-s − 140.·27-s + 276.·29-s − 77.6·30-s − 66.8·31-s + 32.9·32-s + ⋯ |
| L(s) = 1 | + 0.953·2-s + 1.00·3-s − 0.0914·4-s − 0.495·5-s + 0.953·6-s − 1.04·8-s + 0.000930·9-s − 0.472·10-s + 0.700·11-s − 0.0914·12-s − 1.20·13-s − 0.495·15-s − 0.900·16-s − 0.315·17-s + 0.000887·18-s − 0.0123·19-s + 0.0453·20-s + 0.667·22-s + 1.09·23-s − 1.04·24-s − 0.754·25-s − 1.14·26-s − 0.999·27-s + 1.77·29-s − 0.472·30-s − 0.387·31-s + 0.182·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.378994596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.378994596\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 - 2.69T + 8T^{2} \) |
| 3 | \( 1 - 5.19T + 27T^{2} \) |
| 5 | \( 1 + 5.54T + 125T^{2} \) |
| 11 | \( 1 - 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.02T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 66.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 17.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 232.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 545.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 7.63T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 175.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 637.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 880.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 737.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 491.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 571.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 539.T + 9.12e5T^{2} \) |
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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
−8.733372583782276199688203276036, −7.906452486954929628107909501693, −7.13927860344684294232511202887, −6.23633343211424057982034687965, −5.28717159636239524008641092681, −4.44907784659301411092599916366, −3.84928700435952430662997579213, −2.95003899993456914457280104927, −2.33601027038679079913551226621, −0.65438719546778774221008046022,
0.65438719546778774221008046022, 2.33601027038679079913551226621, 2.95003899993456914457280104927, 3.84928700435952430662997579213, 4.44907784659301411092599916366, 5.28717159636239524008641092681, 6.23633343211424057982034687965, 7.13927860344684294232511202887, 7.906452486954929628107909501693, 8.733372583782276199688203276036
