L(s) = 1 | − 0.414·2-s + 9.24·3-s − 7.82·4-s + 0.656·5-s − 3.82·6-s + 6.14·7-s + 6.55·8-s + 58.4·9-s − 0.272·10-s + 65.3·11-s − 72.3·12-s − 49.7·13-s − 2.54·14-s + 6.07·15-s + 59.9·16-s − 55.4·17-s − 24.2·18-s + 64.7·19-s − 5.14·20-s + 56.7·21-s − 27.0·22-s + 93.8·23-s + 60.5·24-s − 124.·25-s + 20.6·26-s + 290.·27-s − 48.0·28-s + ⋯ |
L(s) = 1 | − 0.146·2-s + 1.77·3-s − 0.978·4-s + 0.0587·5-s − 0.260·6-s + 0.331·7-s + 0.289·8-s + 2.16·9-s − 0.00860·10-s + 1.79·11-s − 1.74·12-s − 1.06·13-s − 0.0485·14-s + 0.104·15-s + 0.936·16-s − 0.791·17-s − 0.316·18-s + 0.781·19-s − 0.0574·20-s + 0.589·21-s − 0.262·22-s + 0.851·23-s + 0.515·24-s − 0.996·25-s + 0.155·26-s + 2.07·27-s − 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.525045624\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.525045624\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + 0.414T + 8T^{2} \) |
| 3 | \( 1 - 9.24T + 27T^{2} \) |
| 5 | \( 1 - 0.656T + 125T^{2} \) |
| 7 | \( 1 - 6.14T + 343T^{2} \) |
| 11 | \( 1 - 65.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 64.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.8T + 1.21e4T^{2} \) |
| 31 | \( 1 - 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 76.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 80.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 844.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 378.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 353.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 696.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 805.T + 9.12e5T^{2} \) |
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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
−9.589501026350119493705285476551, −8.941113540689246103215420912440, −8.388307113500313299338703885107, −7.50768133227673340277756820612, −6.68311863926631017039560372448, −5.01265273433794332958452392924, −4.16371286343589012715517510469, −3.44775024398054508960177529318, −2.19865151188190038502223309421, −1.06831714854413297563034374258,
1.06831714854413297563034374258, 2.19865151188190038502223309421, 3.44775024398054508960177529318, 4.16371286343589012715517510469, 5.01265273433794332958452392924, 6.68311863926631017039560372448, 7.50768133227673340277756820612, 8.388307113500313299338703885107, 8.941113540689246103215420912440, 9.589501026350119493705285476551