| L(s) = 1 | + 0.633·2-s − 1.59·4-s − 4.04·5-s − 3.23·7-s − 2.27·8-s − 2.55·10-s + 11-s + 13-s − 2.04·14-s + 1.75·16-s − 1.26·17-s − 3.76·19-s + 6.46·20-s + 0.633·22-s + 4.00·23-s + 11.3·25-s + 0.633·26-s + 5.17·28-s − 2.63·29-s − 3.14·31-s + 5.66·32-s − 0.801·34-s + 13.0·35-s − 8.50·37-s − 2.38·38-s + 9.21·40-s + 10.4·41-s + ⋯ |
| L(s) = 1 | + 0.447·2-s − 0.799·4-s − 1.80·5-s − 1.22·7-s − 0.805·8-s − 0.809·10-s + 0.301·11-s + 0.277·13-s − 0.547·14-s + 0.439·16-s − 0.307·17-s − 0.863·19-s + 1.44·20-s + 0.134·22-s + 0.834·23-s + 2.26·25-s + 0.124·26-s + 0.978·28-s − 0.490·29-s − 0.564·31-s + 1.00·32-s − 0.137·34-s + 2.21·35-s − 1.39·37-s − 0.386·38-s + 1.45·40-s + 1.62·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6046779093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6046779093\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.633T + 2T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14T + 31T^{2} \) |
| 37 | \( 1 + 8.50T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 - 3.82T + 47T^{2} \) |
| 53 | \( 1 + 0.485T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 1.58T + 73T^{2} \) |
| 79 | \( 1 - 4.63T + 79T^{2} \) |
| 83 | \( 1 + 9.85T + 83T^{2} \) |
| 89 | \( 1 - 6.51T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−9.395503525509788681546944150370, −8.896961625077819475229112281041, −8.104919505801533270796432640337, −7.16237873070519967576990894468, −6.41445104575625123408244370104, −5.29482209324596650688706833037, −4.18144442116717768894644706712, −3.80133361877218163113457336364, −2.94571841820981767046666092620, −0.51682816302529156252111322005,
0.51682816302529156252111322005, 2.94571841820981767046666092620, 3.80133361877218163113457336364, 4.18144442116717768894644706712, 5.29482209324596650688706833037, 6.41445104575625123408244370104, 7.16237873070519967576990894468, 8.104919505801533270796432640337, 8.896961625077819475229112281041, 9.395503525509788681546944150370
