| L(s) = 1 | − 2.48·2-s − 3-s + 4.17·4-s − 4.10·5-s + 2.48·6-s + 7-s − 5.39·8-s + 9-s + 10.2·10-s + 4.28·11-s − 4.17·12-s − 3.43·13-s − 2.48·14-s + 4.10·15-s + 5.06·16-s − 4.49·17-s − 2.48·18-s − 2.97·19-s − 17.1·20-s − 21-s − 10.6·22-s − 5.23·23-s + 5.39·24-s + 11.8·25-s + 8.52·26-s − 27-s + 4.17·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 0.577·3-s + 2.08·4-s − 1.83·5-s + 1.01·6-s + 0.377·7-s − 1.90·8-s + 0.333·9-s + 3.22·10-s + 1.29·11-s − 1.20·12-s − 0.951·13-s − 0.663·14-s + 1.06·15-s + 1.26·16-s − 1.09·17-s − 0.585·18-s − 0.682·19-s − 3.83·20-s − 0.218·21-s − 2.26·22-s − 1.09·23-s + 1.10·24-s + 2.37·25-s + 1.67·26-s − 0.192·27-s + 0.788·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2493458616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2493458616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 2.97T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 0.906T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 6.04T + 37T^{2} \) |
| 43 | \( 1 + 1.10T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 4.69T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 8.03T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 2.08T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 - 3.38T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 0.905T + 89T^{2} \) |
| 97 | \( 1 + 2.06T + 97T^{2} \) |
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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
−10.16408043926214968095011587882, −9.173477551753848677902527070757, −8.487403538485041808189638818976, −7.74445896579224566819031912726, −7.07277561784216525545871502533, −6.39612052616486561227354819069, −4.66176008421724595225227668399, −3.79751120485570733119824775047, −2.04987959890143345698746845205, −0.52012378538432449797403503038,
0.52012378538432449797403503038, 2.04987959890143345698746845205, 3.79751120485570733119824775047, 4.66176008421724595225227668399, 6.39612052616486561227354819069, 7.07277561784216525545871502533, 7.74445896579224566819031912726, 8.487403538485041808189638818976, 9.173477551753848677902527070757, 10.16408043926214968095011587882
