| L(s) = 1 | − 2.27·2-s − 3·3-s − 2.82·4-s + 6.82·6-s − 7·7-s + 24.6·8-s + 9·9-s − 40.7·11-s + 8.47·12-s − 53.2·13-s + 15.9·14-s − 33.4·16-s − 4.54·17-s − 20.4·18-s + 122.·19-s + 21·21-s + 92.7·22-s − 131.·23-s − 73.8·24-s + 121.·26-s − 27·27-s + 19.7·28-s − 216.·29-s − 251.·31-s − 120.·32-s + 122.·33-s + 10.3·34-s + ⋯ |
| L(s) = 1 | − 0.804·2-s − 0.577·3-s − 0.353·4-s + 0.464·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s − 1.11·11-s + 0.203·12-s − 1.13·13-s + 0.303·14-s − 0.522·16-s − 0.0649·17-s − 0.268·18-s + 1.48·19-s + 0.218·21-s + 0.898·22-s − 1.19·23-s − 0.628·24-s + 0.914·26-s − 0.192·27-s + 0.133·28-s − 1.38·29-s − 1.45·31-s − 0.668·32-s + 0.644·33-s + 0.0522·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3677304989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3677304989\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 2.27T + 8T^{2} \) |
| 11 | \( 1 + 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 637.T + 9.12e5T^{2} \) |
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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
−10.07504274253965074396877376947, −9.841950900499219603784809935192, −8.786788349037375211780137849644, −7.61500552686563996530543626386, −7.24162752800795413716247281132, −5.61722018365470628775424047498, −5.02081637929328262070100018977, −3.68600692427838551194004425840, −2.05253723910564937339151222034, −0.41932831911900778266240636138,
0.41932831911900778266240636138, 2.05253723910564937339151222034, 3.68600692427838551194004425840, 5.02081637929328262070100018977, 5.61722018365470628775424047498, 7.24162752800795413716247281132, 7.61500552686563996530543626386, 8.786788349037375211780137849644, 9.841950900499219603784809935192, 10.07504274253965074396877376947
