| 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
| \(L(1)\) |
\(\approx\) |
\(0.5283517105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5283517105\) |
| \(L(\frac{3}{2})\) |
|
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
−8.179973661375063929673139627961, −7.75259560555948641402045641659, −7.37186909254632687565611469841, −6.26172620424009590100880126171, −5.50627003545573358916860676169, −4.69922555728763204763305805093, −3.62155193860924093862598511061, −3.29805567501927992257675175863, −2.10768653893582006582456553436, −0.39758180144335461761380036596,
0.39758180144335461761380036596, 2.10768653893582006582456553436, 3.29805567501927992257675175863, 3.62155193860924093862598511061, 4.69922555728763204763305805093, 5.50627003545573358916860676169, 6.26172620424009590100880126171, 7.37186909254632687565611469841, 7.75259560555948641402045641659, 8.179973661375063929673139627961
