| L(s) = 1 | + 3.35·3-s + 0.258·5-s − 2.55·7-s + 8.26·9-s + 0.430·11-s + 2.67·13-s + 0.867·15-s − 1.88·17-s − 8.10·19-s − 8.59·21-s + 2.92·23-s − 4.93·25-s + 17.6·27-s − 0.559·29-s + 7.15·31-s + 1.44·33-s − 0.661·35-s + 10.8·37-s + 8.97·39-s + 8.47·41-s − 0.721·43-s + 2.13·45-s + 8.40·47-s − 0.449·49-s − 6.31·51-s − 1.64·53-s + 0.111·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s + 0.115·5-s − 0.967·7-s + 2.75·9-s + 0.129·11-s + 0.742·13-s + 0.223·15-s − 0.456·17-s − 1.86·19-s − 1.87·21-s + 0.610·23-s − 0.986·25-s + 3.40·27-s − 0.103·29-s + 1.28·31-s + 0.251·33-s − 0.111·35-s + 1.77·37-s + 1.43·39-s + 1.32·41-s − 0.109·43-s + 0.318·45-s + 1.22·47-s − 0.0641·49-s − 0.884·51-s − 0.225·53-s + 0.0149·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.414488071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.414488071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 3 | \( 1 - 3.35T + 3T^{2} \) |
| 5 | \( 1 - 0.258T + 5T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 11 | \( 1 - 0.430T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 + 8.10T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 + 0.559T + 29T^{2} \) |
| 31 | \( 1 - 7.15T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 + 0.721T + 43T^{2} \) |
| 47 | \( 1 - 8.40T + 47T^{2} \) |
| 53 | \( 1 + 1.64T + 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 4.59T + 67T^{2} \) |
| 71 | \( 1 + 9.22T + 71T^{2} \) |
| 73 | \( 1 - 0.224T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 + 9.34T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 1.15T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.941167873578969330035184374783, −7.19203585581796176530628140191, −6.47746981064793276821193699451, −6.01228908718362206958275533945, −4.49545354799175695295027478221, −4.10121658290831738038202708114, −3.41318293404087264346002705564, −2.56438858285312690802533021804, −2.16045547468644830196980864764, −0.938883504984819081264084076825,
0.938883504984819081264084076825, 2.16045547468644830196980864764, 2.56438858285312690802533021804, 3.41318293404087264346002705564, 4.10121658290831738038202708114, 4.49545354799175695295027478221, 6.01228908718362206958275533945, 6.47746981064793276821193699451, 7.19203585581796176530628140191, 7.941167873578969330035184374783
