L(s) = 1 | − 0.395·2-s − 1.84·4-s + 0.579·5-s − 2.59·7-s + 1.52·8-s − 0.229·10-s − 1.87·11-s − 6.20·13-s + 1.02·14-s + 3.08·16-s − 4.01·17-s − 2.79·19-s − 1.06·20-s + 0.742·22-s − 23-s − 4.66·25-s + 2.45·26-s + 4.78·28-s + 29-s − 3.08·31-s − 4.26·32-s + 1.58·34-s − 1.50·35-s − 3.22·37-s + 1.10·38-s + 0.881·40-s + 8.40·41-s + ⋯ |
L(s) = 1 | − 0.279·2-s − 0.921·4-s + 0.259·5-s − 0.981·7-s + 0.537·8-s − 0.0725·10-s − 0.565·11-s − 1.72·13-s + 0.274·14-s + 0.770·16-s − 0.973·17-s − 0.640·19-s − 0.238·20-s + 0.158·22-s − 0.208·23-s − 0.932·25-s + 0.482·26-s + 0.904·28-s + 0.185·29-s − 0.554·31-s − 0.753·32-s + 0.272·34-s − 0.254·35-s − 0.530·37-s + 0.179·38-s + 0.139·40-s + 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2165730882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2165730882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 0.395T + 2T^{2} \) |
| 5 | \( 1 - 0.579T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 + 6.56T + 43T^{2} \) |
| 47 | \( 1 - 0.0783T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 + 9.00T + 61T^{2} \) |
| 67 | \( 1 - 0.596T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 - 0.703T + 73T^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 - 9.68T + 83T^{2} \) |
| 89 | \( 1 + 9.96T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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
−8.045882132446684994918831957160, −7.50574971498764225822465494976, −6.69493914342626017212843749091, −5.95087352229869523115834687288, −5.09469104310905038067243384202, −4.54376156080348832955329916436, −3.69509683692094053840668765420, −2.72138382605600410172019710103, −1.90662019550431645480988711097, −0.24156731249898438173100670306,
0.24156731249898438173100670306, 1.90662019550431645480988711097, 2.72138382605600410172019710103, 3.69509683692094053840668765420, 4.54376156080348832955329916436, 5.09469104310905038067243384202, 5.95087352229869523115834687288, 6.69493914342626017212843749091, 7.50574971498764225822465494976, 8.045882132446684994918831957160