| L(s) = 1 | − 3·3-s − 2.77·5-s − 5.23·7-s + 9·9-s − 26.0·11-s + 13·13-s + 8.32·15-s + 4.06·17-s + 73.9·19-s + 15.6·21-s − 145.·23-s − 117.·25-s − 27·27-s + 259.·29-s + 78.5·31-s + 78.0·33-s + 14.5·35-s − 99.8·37-s − 39·39-s + 346.·41-s + 137.·43-s − 24.9·45-s + 56.7·47-s − 315.·49-s − 12.1·51-s − 173.·53-s + 72.2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.248·5-s − 0.282·7-s + 0.333·9-s − 0.713·11-s + 0.277·13-s + 0.143·15-s + 0.0579·17-s + 0.893·19-s + 0.163·21-s − 1.31·23-s − 0.938·25-s − 0.192·27-s + 1.66·29-s + 0.455·31-s + 0.411·33-s + 0.0701·35-s − 0.443·37-s − 0.160·39-s + 1.31·41-s + 0.488·43-s − 0.0827·45-s + 0.176·47-s − 0.920·49-s − 0.0334·51-s − 0.449·53-s + 0.177·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 2.77T + 125T^{2} \) |
| 7 | \( 1 + 5.23T + 343T^{2} \) |
| 11 | \( 1 + 26.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 4.06T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 78.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 346.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 56.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 173.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 675.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 798.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 292.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 670.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 786.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 317.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 886.T + 9.12e5T^{2} \) |
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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
−7.982478270124590159127077990747, −7.57234548875484384333073274301, −6.48154264950208503416041401362, −5.93348875997616272664582114323, −5.07026924190639187302453605560, −4.24931374690436510704310812883, −3.31962697441669604665039528626, −2.29463016680673451048576056544, −1.02408165677062171441291786225, 0,
1.02408165677062171441291786225, 2.29463016680673451048576056544, 3.31962697441669604665039528626, 4.24931374690436510704310812883, 5.07026924190639187302453605560, 5.93348875997616272664582114323, 6.48154264950208503416041401362, 7.57234548875484384333073274301, 7.982478270124590159127077990747
