| L(s) = 1 | − 2.82·3-s + 0.0936·5-s − 0.650·7-s + 4.96·9-s + 5.87·11-s + 5.73·13-s − 0.264·15-s + 2.88·17-s + 19-s + 1.83·21-s + 7.57·23-s − 4.99·25-s − 5.55·27-s + 0.702·29-s + 10.9·31-s − 16.5·33-s − 0.0608·35-s + 0.728·37-s − 16.1·39-s + 1.10·41-s − 1.59·43-s + 0.465·45-s + 10.1·47-s − 6.57·49-s − 8.14·51-s + 53-s + 0.550·55-s + ⋯ |
| L(s) = 1 | − 1.62·3-s + 0.0418·5-s − 0.245·7-s + 1.65·9-s + 1.77·11-s + 1.59·13-s − 0.0682·15-s + 0.700·17-s + 0.229·19-s + 0.400·21-s + 1.57·23-s − 0.998·25-s − 1.06·27-s + 0.130·29-s + 1.96·31-s − 2.88·33-s − 0.0102·35-s + 0.119·37-s − 2.59·39-s + 0.171·41-s − 0.242·43-s + 0.0693·45-s + 1.47·47-s − 0.939·49-s − 1.14·51-s + 0.137·53-s + 0.0742·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477137415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477137415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.0936T + 5T^{2} \) |
| 7 | \( 1 + 0.650T + 7T^{2} \) |
| 11 | \( 1 - 5.87T + 11T^{2} \) |
| 13 | \( 1 - 5.73T + 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 - 0.702T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 0.728T + 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 3.05T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 2.39T + 79T^{2} \) |
| 83 | \( 1 - 3.94T + 83T^{2} \) |
| 89 | \( 1 - 9.35T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.564579111618950079624431640902, −7.45926266228170188573284228105, −6.65312619327582924778212181443, −6.19095738474440843344132405666, −5.73805141816129647060554973275, −4.71411054797396560942322374759, −4.03618106793026381254153563578, −3.17239278514501999455921366455, −1.37180045620027547255331447104, −0.909686610341033514736384436541,
0.909686610341033514736384436541, 1.37180045620027547255331447104, 3.17239278514501999455921366455, 4.03618106793026381254153563578, 4.71411054797396560942322374759, 5.73805141816129647060554973275, 6.19095738474440843344132405666, 6.65312619327582924778212181443, 7.45926266228170188573284228105, 8.564579111618950079624431640902
