| L(s) = 1 | − 8.38e6·2-s + 2.72e11·3-s + 7.03e13·4-s + 3.64e16·5-s − 2.28e18·6-s − 2.73e19·7-s − 5.90e20·8-s + 4.74e22·9-s − 3.05e23·10-s − 2.09e24·11-s + 1.91e25·12-s + 1.03e26·13-s + 2.29e26·14-s + 9.90e27·15-s + 4.95e27·16-s + 1.00e29·17-s − 3.97e29·18-s − 1.09e30·19-s + 2.56e30·20-s − 7.44e30·21-s + 1.75e31·22-s + 1.84e31·23-s − 1.60e32·24-s + 6.15e32·25-s − 8.64e32·26-s + 5.67e33·27-s − 1.92e33·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s + 1.36·5-s − 1.17·6-s − 0.377·7-s − 0.353·8-s + 1.78·9-s − 0.966·10-s − 0.705·11-s + 0.834·12-s + 0.684·13-s + 0.267·14-s + 2.27·15-s + 0.250·16-s + 1.22·17-s − 1.26·18-s − 0.972·19-s + 0.683·20-s − 0.630·21-s + 0.498·22-s + 0.184·23-s − 0.589·24-s + 0.866·25-s − 0.484·26-s + 1.30·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(48-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+47/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(24)\) |
\(\approx\) |
\(4.560338986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.560338986\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.38e6T \) |
| 7 | \( 1 + 2.73e19T \) |
| good | 3 | \( 1 - 2.72e11T + 2.65e22T^{2} \) |
| 5 | \( 1 - 3.64e16T + 7.10e32T^{2} \) |
| 11 | \( 1 + 2.09e24T + 8.81e48T^{2} \) |
| 13 | \( 1 - 1.03e26T + 2.26e52T^{2} \) |
| 17 | \( 1 - 1.00e29T + 6.77e57T^{2} \) |
| 19 | \( 1 + 1.09e30T + 1.26e60T^{2} \) |
| 23 | \( 1 - 1.84e31T + 1.00e64T^{2} \) |
| 29 | \( 1 - 2.92e34T + 5.40e68T^{2} \) |
| 31 | \( 1 + 1.21e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 3.53e35T + 5.07e73T^{2} \) |
| 41 | \( 1 - 3.31e37T + 6.32e75T^{2} \) |
| 43 | \( 1 + 3.35e38T + 5.92e76T^{2} \) |
| 47 | \( 1 - 3.64e39T + 3.87e78T^{2} \) |
| 53 | \( 1 - 3.58e40T + 1.09e81T^{2} \) |
| 59 | \( 1 + 4.36e40T + 1.69e83T^{2} \) |
| 61 | \( 1 - 1.44e42T + 8.13e83T^{2} \) |
| 67 | \( 1 - 3.41e42T + 6.69e85T^{2} \) |
| 71 | \( 1 - 5.28e43T + 1.02e87T^{2} \) |
| 73 | \( 1 - 4.83e43T + 3.76e87T^{2} \) |
| 79 | \( 1 - 4.04e44T + 1.54e89T^{2} \) |
| 83 | \( 1 + 2.36e45T + 1.57e90T^{2} \) |
| 89 | \( 1 - 5.16e45T + 4.18e91T^{2} \) |
| 97 | \( 1 + 4.02e46T + 2.38e93T^{2} \) |
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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
−10.48731456085658897318927816181, −9.792084120974156545768337626925, −8.876733315877629796074741201833, −8.062244251265192449233883056347, −6.79911697198367824911230277171, −5.54429472712642163667605745911, −3.67461111341954555030019017421, −2.63504434782071486768287740711, −1.99749086842768582306736982959, −0.945290122044336959097392715880,
0.945290122044336959097392715880, 1.99749086842768582306736982959, 2.63504434782071486768287740711, 3.67461111341954555030019017421, 5.54429472712642163667605745911, 6.79911697198367824911230277171, 8.062244251265192449233883056347, 8.876733315877629796074741201833, 9.792084120974156545768337626925, 10.48731456085658897318927816181
