| L(s) = 1 | − 1.82·2-s + 3.09·3-s + 1.32·4-s − 3.15·5-s − 5.64·6-s − 0.287·7-s + 1.23·8-s + 6.60·9-s + 5.74·10-s + 11-s + 4.10·12-s + 0.910·13-s + 0.524·14-s − 9.77·15-s − 4.89·16-s + 0.621·17-s − 12.0·18-s + 6.81·19-s − 4.17·20-s − 0.890·21-s − 1.82·22-s − 7.16·23-s + 3.82·24-s + 4.94·25-s − 1.66·26-s + 11.1·27-s − 0.380·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.78·3-s + 0.661·4-s − 1.41·5-s − 2.30·6-s − 0.108·7-s + 0.436·8-s + 2.20·9-s + 1.81·10-s + 0.301·11-s + 1.18·12-s + 0.252·13-s + 0.140·14-s − 2.52·15-s − 1.22·16-s + 0.150·17-s − 2.83·18-s + 1.56·19-s − 0.933·20-s − 0.194·21-s − 0.388·22-s − 1.49·23-s + 0.780·24-s + 0.989·25-s − 0.325·26-s + 2.15·27-s − 0.0718·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177916224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177916224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 3.09T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 0.287T + 7T^{2} \) |
| 13 | \( 1 - 0.910T + 13T^{2} \) |
| 17 | \( 1 - 0.621T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 + 7.16T + 23T^{2} \) |
| 29 | \( 1 - 7.26T + 29T^{2} \) |
| 31 | \( 1 - 5.73T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 - 1.62T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 - 4.08T + 59T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 - 9.71T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 + 8.93T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 3.68T + 97T^{2} \) |
| 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
−9.840106266440173777865960355265, −9.679960666425160282546182261000, −8.540035805720716481301599058108, −8.029826311928305858095110919174, −7.70678186425726683884107186190, −6.70535691258700368117708877799, −4.50336507884121581204348000186, −3.72307827168967374326251090004, −2.66673304094879095611165429930, −1.11325866548542775683209569927,
1.11325866548542775683209569927, 2.66673304094879095611165429930, 3.72307827168967374326251090004, 4.50336507884121581204348000186, 6.70535691258700368117708877799, 7.70678186425726683884107186190, 8.029826311928305858095110919174, 8.540035805720716481301599058108, 9.679960666425160282546182261000, 9.840106266440173777865960355265
