| L(s) = 1 | + 0.844·5-s − 5.06·7-s − 2.84·11-s + 2.90·13-s − 7.72·17-s + 19-s − 3.91·23-s − 4.28·25-s − 2.90·29-s − 7.00·31-s − 4.27·35-s + 9.00·37-s + 6.81·41-s + 5.96·43-s + 1.04·47-s + 18.6·49-s + 9.03·53-s − 2.40·55-s − 0.749·59-s + 2.27·61-s + 2.45·65-s + 10.3·67-s − 2.13·71-s + 4.60·73-s + 14.4·77-s + 3.88·79-s − 8.44·83-s + ⋯ |
| L(s) = 1 | + 0.377·5-s − 1.91·7-s − 0.857·11-s + 0.804·13-s − 1.87·17-s + 0.229·19-s − 0.815·23-s − 0.857·25-s − 0.538·29-s − 1.25·31-s − 0.723·35-s + 1.48·37-s + 1.06·41-s + 0.910·43-s + 0.151·47-s + 2.66·49-s + 1.24·53-s − 0.323·55-s − 0.0975·59-s + 0.291·61-s + 0.304·65-s + 1.26·67-s − 0.252·71-s + 0.538·73-s + 1.64·77-s + 0.437·79-s − 0.926·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9153372847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9153372847\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 0.844T + 5T^{2} \) |
| 7 | \( 1 + 5.06T + 7T^{2} \) |
| 11 | \( 1 + 2.84T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 + 7.72T + 17T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 - 9.00T + 37T^{2} \) |
| 41 | \( 1 - 6.81T + 41T^{2} \) |
| 43 | \( 1 - 5.96T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 - 9.03T + 53T^{2} \) |
| 59 | \( 1 + 0.749T + 59T^{2} \) |
| 61 | \( 1 - 2.27T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.196203216737642214574798704879, −7.30128408852210830322434367304, −6.69107350028288873714971514060, −5.89554309957353020069065872274, −5.67756115309310424932057230418, −4.24316500237545842419887717422, −3.76189749426023620881916728087, −2.71565441017914983101763142581, −2.13684230202597307160489410611, −0.47825230995625453088936185001,
0.47825230995625453088936185001, 2.13684230202597307160489410611, 2.71565441017914983101763142581, 3.76189749426023620881916728087, 4.24316500237545842419887717422, 5.67756115309310424932057230418, 5.89554309957353020069065872274, 6.69107350028288873714971514060, 7.30128408852210830322434367304, 8.196203216737642214574798704879
