| L(s) = 1 | − 3·5-s + 2.23·7-s + 4.47·11-s − 13-s + 3·17-s − 4.47·19-s + 8.94·23-s + 4·25-s − 10·29-s − 6.70·35-s + 3·37-s + 6.70·43-s − 2.23·47-s − 1.99·49-s − 4·53-s − 13.4·55-s + 4.47·59-s + 3·65-s + 13.4·67-s − 6.70·71-s + 14·73-s + 10.0·77-s − 8.94·79-s − 17.8·83-s − 9·85-s + 10·89-s − 2.23·91-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.845·7-s + 1.34·11-s − 0.277·13-s + 0.727·17-s − 1.02·19-s + 1.86·23-s + 0.800·25-s − 1.85·29-s − 1.13·35-s + 0.493·37-s + 1.02·43-s − 0.326·47-s − 0.285·49-s − 0.549·53-s − 1.80·55-s + 0.582·59-s + 0.372·65-s + 1.63·67-s − 0.796·71-s + 1.63·73-s + 1.13·77-s − 1.00·79-s − 1.96·83-s − 0.976·85-s + 1.05·89-s − 0.234·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671909818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671909818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 4.47T + 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 6.70T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.515245810440513603401246871378, −7.67874062083405697403242203637, −7.25744860291917568378418205894, −6.42621022266829406806621269248, −5.39207713387039701470231963984, −4.54966527534769299636854543770, −3.95102930778469607942271498610, −3.21486104860728289780686034821, −1.86810220297277951543526585012, −0.77836902662681759368104591465,
0.77836902662681759368104591465, 1.86810220297277951543526585012, 3.21486104860728289780686034821, 3.95102930778469607942271498610, 4.54966527534769299636854543770, 5.39207713387039701470231963984, 6.42621022266829406806621269248, 7.25744860291917568378418205894, 7.67874062083405697403242203637, 8.515245810440513603401246871378
