| L(s) = 1 | + 9.97·5-s + 13.8·7-s + 8.29·11-s + 28.2·17-s − 19·19-s − 34.8·23-s + 74.4·25-s + 137.·35-s − 53.8·43-s + 36.6·47-s + 142.·49-s + 82.7·55-s − 5.12·61-s − 112.·73-s + 114.·77-s − 139.·83-s + 281.·85-s − 189.·95-s − 174.·101-s − 347.·115-s + 390.·119-s + ⋯ |
| L(s) = 1 | + 1.99·5-s + 1.97·7-s + 0.754·11-s + 1.66·17-s − 19-s − 1.51·23-s + 2.97·25-s + 3.93·35-s − 1.25·43-s + 0.779·47-s + 2.90·49-s + 1.50·55-s − 0.0839·61-s − 1.53·73-s + 1.49·77-s − 1.68·83-s + 3.31·85-s − 1.99·95-s − 1.72·101-s − 3.02·115-s + 3.28·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.656669558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.656669558\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 - 9.97T + 25T^{2} \) |
| 7 | \( 1 - 13.8T + 49T^{2} \) |
| 11 | \( 1 - 8.29T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 28.2T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 36.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.12T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 112.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−8.617809007193918357128923747483, −8.069677777259786567353722045824, −7.09452231726531022878801438809, −6.10398200251447267604506751106, −5.61251089964996718475638044025, −4.92474548759582639590171236598, −4.02866273848910097717252553513, −2.57692880517192781379660765564, −1.69233276144741908968396916920, −1.31248494290328429058399405040,
1.31248494290328429058399405040, 1.69233276144741908968396916920, 2.57692880517192781379660765564, 4.02866273848910097717252553513, 4.92474548759582639590171236598, 5.61251089964996718475638044025, 6.10398200251447267604506751106, 7.09452231726531022878801438809, 8.069677777259786567353722045824, 8.617809007193918357128923747483
