| L(s) = 1 | + 2-s + 4-s + 2.70·5-s + 7-s + 8-s + 2.70·10-s + 4·11-s − 0.701·13-s + 14-s + 16-s − 4·17-s + 7.40·19-s + 2.70·20-s + 4·22-s − 23-s + 2.29·25-s − 0.701·26-s + 28-s − 6.70·29-s − 2·31-s + 32-s − 4·34-s + 2.70·35-s + 10.7·37-s + 7.40·38-s + 2.70·40-s + 6.70·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.20·5-s + 0.377·7-s + 0.353·8-s + 0.854·10-s + 1.20·11-s − 0.194·13-s + 0.267·14-s + 0.250·16-s − 0.970·17-s + 1.69·19-s + 0.604·20-s + 0.852·22-s − 0.208·23-s + 0.459·25-s − 0.137·26-s + 0.188·28-s − 1.24·29-s − 0.359·31-s + 0.176·32-s − 0.685·34-s + 0.456·35-s + 1.75·37-s + 1.20·38-s + 0.427·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.181587461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.181587461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2.70T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 0.701T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 7.40T + 19T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 5.40T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 0.596T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−9.126757857835755877839062312135, −7.77559038018779645695992737940, −7.17492355305916466483826233031, −6.14073453144146507667082244266, −5.86800538294389149040174988243, −4.87896623063302869473157119088, −4.15045559389074793520254337776, −3.10296351108977347968222332467, −2.09490712769327407997606593046, −1.29393395565374479541145938594,
1.29393395565374479541145938594, 2.09490712769327407997606593046, 3.10296351108977347968222332467, 4.15045559389074793520254337776, 4.87896623063302869473157119088, 5.86800538294389149040174988243, 6.14073453144146507667082244266, 7.17492355305916466483826233031, 7.77559038018779645695992737940, 9.126757857835755877839062312135
