| L(s) = 1 | + 2.55·2-s + 4.50·4-s − 1.71·5-s + 6.39·8-s − 4.36·10-s + 0.945·11-s − 13-s + 7.30·16-s + 0.170·17-s − 2.87·19-s − 7.71·20-s + 2.41·22-s + 5.66·23-s − 2.06·25-s − 2.55·26-s + 9.22·29-s + 7.36·31-s + 5.83·32-s + 0.435·34-s + 5.50·37-s − 7.33·38-s − 10.9·40-s + 11.0·41-s + 2.58·43-s + 4.26·44-s + 14.4·46-s − 0.418·47-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.25·4-s − 0.765·5-s + 2.26·8-s − 1.38·10-s + 0.285·11-s − 0.277·13-s + 1.82·16-s + 0.0413·17-s − 0.659·19-s − 1.72·20-s + 0.514·22-s + 1.18·23-s − 0.413·25-s − 0.500·26-s + 1.71·29-s + 1.32·31-s + 1.03·32-s + 0.0746·34-s + 0.905·37-s − 1.18·38-s − 1.73·40-s + 1.71·41-s + 0.394·43-s + 0.642·44-s + 2.13·46-s − 0.0610·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.873371249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.873371249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 1.71T + 5T^{2} \) |
| 11 | \( 1 - 0.945T + 11T^{2} \) |
| 17 | \( 1 - 0.170T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 - 5.66T + 23T^{2} \) |
| 29 | \( 1 - 9.22T + 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 - 5.50T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 + 0.418T + 47T^{2} \) |
| 53 | \( 1 + 0.626T + 53T^{2} \) |
| 59 | \( 1 - 6.73T + 59T^{2} \) |
| 61 | \( 1 - 8.16T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 - 0.382T + 71T^{2} \) |
| 73 | \( 1 - 2.22T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
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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
−7.87812380709331020418353591132, −7.12227331377230958539559996929, −6.52485327324388217365399261163, −5.90995629014393130397524467085, −5.00719805742073154640283121268, −4.38643153326699797856888483591, −3.94844897815377749384749863894, −2.93532147310390676683493048817, −2.44503630035369724394998096790, −1.00869950985511418367298173015,
1.00869950985511418367298173015, 2.44503630035369724394998096790, 2.93532147310390676683493048817, 3.94844897815377749384749863894, 4.38643153326699797856888483591, 5.00719805742073154640283121268, 5.90995629014393130397524467085, 6.52485327324388217365399261163, 7.12227331377230958539559996929, 7.87812380709331020418353591132
