| L(s) = 1 | + 7-s − 3·11-s − 5·13-s + 3·17-s − 2·19-s + 6·23-s − 3·29-s + 4·31-s − 2·37-s + 12·41-s − 10·43-s − 9·47-s + 49-s + 12·53-s + 8·61-s − 4·67-s − 2·73-s − 3·77-s + 79-s − 12·83-s + 12·89-s − 5·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s − 1.38·13-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 0.557·29-s + 0.718·31-s − 0.328·37-s + 1.87·41-s − 1.52·43-s − 1.31·47-s + 1/7·49-s + 1.64·53-s + 1.02·61-s − 0.488·67-s − 0.234·73-s − 0.341·77-s + 0.112·79-s − 1.31·83-s + 1.27·89-s − 0.524·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633418113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633418113\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.16948026538744, −14.84546660163484, −14.46418275446968, −13.75656305585832, −13.01772771928668, −12.85120427098183, −12.10313267096716, −11.58615329015504, −11.07419200230179, −10.22392507156937, −10.14205733920058, −9.328253216627123, −8.745892417642117, −8.041709351011833, −7.609671816180865, −7.066596918043708, −6.422026442864954, −5.478989364024287, −5.158201832164046, −4.580727044941540, −3.773052595184903, −2.853862179976577, −2.464949668978444, −1.521017786445178, −0.5040556459025151,
0.5040556459025151, 1.521017786445178, 2.464949668978444, 2.853862179976577, 3.773052595184903, 4.580727044941540, 5.158201832164046, 5.478989364024287, 6.422026442864954, 7.066596918043708, 7.609671816180865, 8.041709351011833, 8.745892417642117, 9.328253216627123, 10.14205733920058, 10.22392507156937, 11.07419200230179, 11.58615329015504, 12.10313267096716, 12.85120427098183, 13.01772771928668, 13.75656305585832, 14.46418275446968, 14.84546660163484, 15.16948026538744
