L(s) = 1 | − 1.55·2-s + 2.82·3-s + 0.422·4-s + 3.70·5-s − 4.40·6-s + 4.04·7-s + 2.45·8-s + 5.00·9-s − 5.77·10-s − 2.35·11-s + 1.19·12-s + 1.53·13-s − 6.30·14-s + 10.4·15-s − 4.66·16-s − 2.78·17-s − 7.79·18-s − 4.76·19-s + 1.56·20-s + 11.4·21-s + 3.66·22-s − 0.277·23-s + 6.94·24-s + 8.74·25-s − 2.38·26-s + 5.67·27-s + 1.70·28-s + ⋯ |
L(s) = 1 | − 1.10·2-s + 1.63·3-s + 0.211·4-s + 1.65·5-s − 1.79·6-s + 1.53·7-s + 0.868·8-s + 1.66·9-s − 1.82·10-s − 0.709·11-s + 0.344·12-s + 0.424·13-s − 1.68·14-s + 2.70·15-s − 1.16·16-s − 0.676·17-s − 1.83·18-s − 1.09·19-s + 0.349·20-s + 2.50·21-s + 0.780·22-s − 0.0578·23-s + 1.41·24-s + 1.74·25-s − 0.467·26-s + 1.09·27-s + 0.322·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.803331563\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.803331563\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2671 | \( 1 - T \) |
good | 2 | \( 1 + 1.55T + 2T^{2} \) |
| 3 | \( 1 - 2.82T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 + 2.78T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 + 0.277T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 - 0.979T + 31T^{2} \) |
| 37 | \( 1 + 4.86T + 37T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 + 1.72T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 + 4.23T + 59T^{2} \) |
| 61 | \( 1 - 9.60T + 61T^{2} \) |
| 67 | \( 1 - 0.252T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 6.08T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 3.94T + 97T^{2} \) |
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
−9.018623520662554017438989755341, −8.322120043193833692992575048325, −7.75257559792138679071378785202, −6.97498846212340802528327775373, −5.78774781955149053592604901096, −4.84023861100658786457389957940, −4.06264633691640124712982029195, −2.46529105962707966143996104536, −2.08275945198705900527170316196, −1.32679390841958577337630349546,
1.32679390841958577337630349546, 2.08275945198705900527170316196, 2.46529105962707966143996104536, 4.06264633691640124712982029195, 4.84023861100658786457389957940, 5.78774781955149053592604901096, 6.97498846212340802528327775373, 7.75257559792138679071378785202, 8.322120043193833692992575048325, 9.018623520662554017438989755341