| L(s) = 1 | + 0.217·2-s − 7.95·4-s − 7·7-s − 3.46·8-s + 30.6·11-s − 25.3·13-s − 1.52·14-s + 62.8·16-s + 72.8·17-s + 122.·19-s + 6.65·22-s − 194.·23-s − 5.50·26-s + 55.6·28-s − 48.6·29-s − 288.·31-s + 41.4·32-s + 15.8·34-s − 15.8·37-s + 26.6·38-s − 452.·41-s + 152.·43-s − 243.·44-s − 42.2·46-s − 164.·47-s + 49·49-s + 201.·52-s + ⋯ |
| L(s) = 1 | + 0.0768·2-s − 0.994·4-s − 0.377·7-s − 0.153·8-s + 0.838·11-s − 0.540·13-s − 0.0290·14-s + 0.982·16-s + 1.03·17-s + 1.48·19-s + 0.0644·22-s − 1.76·23-s − 0.0415·26-s + 0.375·28-s − 0.311·29-s − 1.67·31-s + 0.228·32-s + 0.0798·34-s − 0.0703·37-s + 0.113·38-s − 1.72·41-s + 0.541·43-s − 0.834·44-s − 0.135·46-s − 0.510·47-s + 0.142·49-s + 0.536·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.356596727\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356596727\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 0.217T + 8T^{2} \) |
| 11 | \( 1 - 30.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 25.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 194.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 48.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 288.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 15.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 452.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 152.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 164.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 591.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 115.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 605.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 990.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 863.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 965.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 160.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 51.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.49e3T + 9.12e5T^{2} \) |
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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
−9.446664912088562002417258195618, −8.206027114613445034726433185969, −7.63327913957691953223041744499, −6.60001654221928856536008121252, −5.59698628813649866924746296388, −5.03461863036153975519851486251, −3.78925843401562490919397163696, −3.40492488701897093498582880582, −1.78336303657521819236818896339, −0.56940861351376336440560729607,
0.56940861351376336440560729607, 1.78336303657521819236818896339, 3.40492488701897093498582880582, 3.78925843401562490919397163696, 5.03461863036153975519851486251, 5.59698628813649866924746296388, 6.60001654221928856536008121252, 7.63327913957691953223041744499, 8.206027114613445034726433185969, 9.446664912088562002417258195618
