| L(s) = 1 | + 1.98·2-s + 1.92·4-s − 1.21·5-s + 0.393·7-s − 0.151·8-s − 2.41·10-s + 2.49·11-s + 3.30·13-s + 0.778·14-s − 4.14·16-s + 4.30·17-s + 5.37·19-s − 2.34·20-s + 4.93·22-s − 3.00·23-s − 3.51·25-s + 6.54·26-s + 0.756·28-s − 10.0·31-s − 7.91·32-s + 8.51·34-s − 0.479·35-s + 3.64·37-s + 10.6·38-s + 0.184·40-s + 7.82·41-s + 2.34·43-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.961·4-s − 0.545·5-s + 0.148·7-s − 0.0535·8-s − 0.763·10-s + 0.751·11-s + 0.916·13-s + 0.208·14-s − 1.03·16-s + 1.04·17-s + 1.23·19-s − 0.524·20-s + 1.05·22-s − 0.625·23-s − 0.702·25-s + 1.28·26-s + 0.142·28-s − 1.79·31-s − 1.39·32-s + 1.46·34-s − 0.0810·35-s + 0.599·37-s + 1.72·38-s + 0.0292·40-s + 1.22·41-s + 0.358·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.366681907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.366681907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 - 0.393T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 3.00T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 - 5.14T + 67T^{2} \) |
| 71 | \( 1 - 8.09T + 71T^{2} \) |
| 73 | \( 1 - 8.75T + 73T^{2} \) |
| 79 | \( 1 - 7.29T + 79T^{2} \) |
| 83 | \( 1 - 3.74T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.64411055089864600715419611707, −7.14421486907497351053755838795, −6.13548031328538610617570697404, −5.74035886313404988206719733297, −5.09374757071788281398070842012, −4.02775985102421895073270165454, −3.81766501046187426063376212975, −3.12100237287409694071350621978, −2.01210791770773190519884009721, −0.868341767749954934486033379859,
0.868341767749954934486033379859, 2.01210791770773190519884009721, 3.12100237287409694071350621978, 3.81766501046187426063376212975, 4.02775985102421895073270165454, 5.09374757071788281398070842012, 5.74035886313404988206719733297, 6.13548031328538610617570697404, 7.14421486907497351053755838795, 7.64411055089864600715419611707
