| L(s) = 1 | − 185.·2-s + 1.93e4·3-s − 4.89e5·4-s − 1.00e6·5-s − 3.58e6·6-s + 3.38e7·7-s + 1.88e8·8-s − 7.88e8·9-s + 1.85e8·10-s − 5.48e9·11-s − 9.46e9·12-s − 2.57e10·13-s − 6.28e9·14-s − 1.93e10·15-s + 2.21e11·16-s + 1.61e11·17-s + 1.46e11·18-s − 2.84e11·19-s + 4.90e11·20-s + 6.54e11·21-s + 1.01e12·22-s − 6.84e12·23-s + 3.63e12·24-s − 1.80e13·25-s + 4.77e12·26-s − 3.76e13·27-s − 1.65e13·28-s + ⋯ |
| L(s) = 1 | − 0.256·2-s + 0.566·3-s − 0.934·4-s − 0.229·5-s − 0.145·6-s + 0.317·7-s + 0.495·8-s − 0.678·9-s + 0.0587·10-s − 0.701·11-s − 0.529·12-s − 0.672·13-s − 0.0812·14-s − 0.130·15-s + 0.807·16-s + 0.329·17-s + 0.173·18-s − 0.202·19-s + 0.214·20-s + 0.179·21-s + 0.179·22-s − 0.792·23-s + 0.280·24-s − 0.947·25-s + 0.172·26-s − 0.951·27-s − 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.8903349187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8903349187\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 1.11e15T \) |
| good | 2 | \( 1 + 185.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 1.93e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 1.00e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 3.38e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 5.48e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.57e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 1.61e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 2.84e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 6.84e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.18e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.15e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.73e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.61e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 1.25e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 9.51e15T + 5.77e32T^{2} \) |
| 59 | \( 1 + 4.42e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.29e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.41e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 7.28e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 6.12e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 2.32e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.61e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 3.71e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.23e19T + 5.60e37T^{2} \) |
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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
−11.87467452306611962912419100901, −10.43621202463853834634849033042, −9.387606540464069223182945443332, −8.308507055344630648088618202066, −7.65209166471103596861281069419, −5.69694750395498732536250635022, −4.57634105771604385147226184567, −3.34050634825728206296714108879, −2.04563255714123045075303692498, −0.44153795868448786403641826045,
0.44153795868448786403641826045, 2.04563255714123045075303692498, 3.34050634825728206296714108879, 4.57634105771604385147226184567, 5.69694750395498732536250635022, 7.65209166471103596861281069419, 8.308507055344630648088618202066, 9.387606540464069223182945443332, 10.43621202463853834634849033042, 11.87467452306611962912419100901
