| L(s) = 1 | − 1.27·2-s − 0.383·4-s + 1.88·5-s + 0.231·7-s + 3.03·8-s − 2.39·10-s + 3.42·11-s + 5.14·13-s − 0.294·14-s − 3.08·16-s + 7.16·19-s − 0.720·20-s − 4.35·22-s + 2.39·23-s − 1.46·25-s − 6.54·26-s − 0.0886·28-s + 4.71·29-s + 4.64·31-s − 2.13·32-s + 0.434·35-s + 4.22·37-s − 9.10·38-s + 5.69·40-s − 0.157·41-s + 8.98·43-s − 1.31·44-s + ⋯ |
| L(s) = 1 | − 0.899·2-s − 0.191·4-s + 0.840·5-s + 0.0874·7-s + 1.07·8-s − 0.756·10-s + 1.03·11-s + 1.42·13-s − 0.0786·14-s − 0.771·16-s + 1.64·19-s − 0.161·20-s − 0.928·22-s + 0.499·23-s − 0.292·25-s − 1.28·26-s − 0.0167·28-s + 0.875·29-s + 0.834·31-s − 0.377·32-s + 0.0735·35-s + 0.694·37-s − 1.47·38-s + 0.900·40-s − 0.0246·41-s + 1.37·43-s − 0.197·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.958766112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.958766112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.27T + 2T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 - 0.231T + 7T^{2} \) |
| 11 | \( 1 - 3.42T + 11T^{2} \) |
| 13 | \( 1 - 5.14T + 13T^{2} \) |
| 19 | \( 1 - 7.16T + 19T^{2} \) |
| 23 | \( 1 - 2.39T + 23T^{2} \) |
| 29 | \( 1 - 4.71T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 0.157T + 41T^{2} \) |
| 43 | \( 1 - 8.98T + 43T^{2} \) |
| 47 | \( 1 - 8.05T + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 0.781T + 67T^{2} \) |
| 71 | \( 1 - 7.87T + 71T^{2} \) |
| 73 | \( 1 + 2.97T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.026333366279086829012322957482, −7.28948660096004679149976175578, −6.49763002515626273225454666048, −5.89572325298054715445361969969, −5.13678287893999935519568385408, −4.25632478048152554532213125228, −3.55744550609112489434188017805, −2.47558957523101486139112229214, −1.24790513806843664440192310884, −1.04085594698273862303915508024,
1.04085594698273862303915508024, 1.24790513806843664440192310884, 2.47558957523101486139112229214, 3.55744550609112489434188017805, 4.25632478048152554532213125228, 5.13678287893999935519568385408, 5.89572325298054715445361969969, 6.49763002515626273225454666048, 7.28948660096004679149976175578, 8.026333366279086829012322957482
