| L(s) = 1 | − 3·7-s − 11-s + 3·17-s + 3·19-s − 3·23-s − 6·29-s + 10·31-s − 7·37-s − 7·41-s − 4·43-s − 9·47-s + 2·49-s − 12·53-s − 7·59-s + 4·61-s − 12·67-s − 15·71-s − 4·73-s + 3·77-s − 15·79-s + 6·83-s + 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.301·11-s + 0.727·17-s + 0.688·19-s − 0.625·23-s − 1.11·29-s + 1.79·31-s − 1.15·37-s − 1.09·41-s − 0.609·43-s − 1.31·47-s + 2/7·49-s − 1.64·53-s − 0.911·59-s + 0.512·61-s − 1.46·67-s − 1.78·71-s − 0.468·73-s + 0.341·77-s − 1.68·79-s + 0.658·83-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2075811913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2075811913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.33404249523848, −12.91620383950965, −12.22318724763413, −11.95639293639141, −11.54877697508325, −10.82478564298592, −10.27049859319701, −9.941322566247305, −9.592539728795805, −9.062031096666740, −8.344483402762777, −8.067832098601214, −7.353865149939260, −7.007182409617855, −6.290993891787504, −6.045467815872026, −5.390303042346913, −4.847306996112259, −4.291072796395508, −3.428061232777651, −3.224646908790613, −2.727110540007707, −1.731353635663285, −1.311631469749886, −0.1314475959629120,
0.1314475959629120, 1.311631469749886, 1.731353635663285, 2.727110540007707, 3.224646908790613, 3.428061232777651, 4.291072796395508, 4.847306996112259, 5.390303042346913, 6.045467815872026, 6.290993891787504, 7.007182409617855, 7.353865149939260, 8.067832098601214, 8.344483402762777, 9.062031096666740, 9.592539728795805, 9.941322566247305, 10.27049859319701, 10.82478564298592, 11.54877697508325, 11.95639293639141, 12.22318724763413, 12.91620383950965, 13.33404249523848
