| L(s) = 1 | − 2.93·2-s + 0.641·4-s + 15.8·5-s − 32.0·7-s + 21.6·8-s − 46.5·10-s + 50.5·11-s + 44.4·13-s + 94.1·14-s − 68.7·16-s + 94.4·17-s − 55.5·19-s + 10.1·20-s − 148.·22-s − 117.·23-s + 126.·25-s − 130.·26-s − 20.5·28-s − 121.·29-s − 101.·31-s + 28.9·32-s − 277.·34-s − 507.·35-s − 143.·37-s + 163.·38-s + 342.·40-s + 248.·41-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.0801·4-s + 1.41·5-s − 1.72·7-s + 0.956·8-s − 1.47·10-s + 1.38·11-s + 0.948·13-s + 1.79·14-s − 1.07·16-s + 1.34·17-s − 0.670·19-s + 0.113·20-s − 1.43·22-s − 1.06·23-s + 1.00·25-s − 0.985·26-s − 0.138·28-s − 0.776·29-s − 0.585·31-s + 0.159·32-s − 1.39·34-s − 2.44·35-s − 0.638·37-s + 0.697·38-s + 1.35·40-s + 0.947·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 + 2.93T + 8T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 + 32.0T + 343T^{2} \) |
| 11 | \( 1 - 50.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 94.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 117.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 143.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 460.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 750.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 750.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 18.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 795.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 607.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 617.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 120.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 410.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 727.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.701330233757849074814240993575, −7.65855323006684096322752105445, −6.58234410771957919498360237145, −6.25193764734675833254390119433, −5.47418659635193191071735128940, −4.01469376287649849745971603680, −3.32483746367753414119147332741, −1.92202066732424800372764346546, −1.20736536006818310924014923666, 0,
1.20736536006818310924014923666, 1.92202066732424800372764346546, 3.32483746367753414119147332741, 4.01469376287649849745971603680, 5.47418659635193191071735128940, 6.25193764734675833254390119433, 6.58234410771957919498360237145, 7.65855323006684096322752105445, 8.701330233757849074814240993575
