| L(s) = 1 | + 1.56·2-s − 2.56·3-s + 0.438·4-s + 5-s − 4·6-s − 7-s − 2.43·8-s + 3.56·9-s + 1.56·10-s + 2.56·11-s − 1.12·12-s + 4.56·13-s − 1.56·14-s − 2.56·15-s − 4.68·16-s − 4.56·17-s + 5.56·18-s + 1.12·19-s + 0.438·20-s + 2.56·21-s + 4·22-s − 5.12·23-s + 6.24·24-s + 25-s + 7.12·26-s − 1.43·27-s − 0.438·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 1.47·3-s + 0.219·4-s + 0.447·5-s − 1.63·6-s − 0.377·7-s − 0.862·8-s + 1.18·9-s + 0.493·10-s + 0.772·11-s − 0.324·12-s + 1.26·13-s − 0.417·14-s − 0.661·15-s − 1.17·16-s − 1.10·17-s + 1.31·18-s + 0.257·19-s + 0.0980·20-s + 0.558·21-s + 0.852·22-s − 1.06·23-s + 1.27·24-s + 0.200·25-s + 1.39·26-s − 0.276·27-s − 0.0828·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8101846184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8101846184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 3.12T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−16.54043015310824634377959576393, −15.47778975023287695392669568729, −13.94588673316197709173832397868, −12.99212700705068834617634779508, −11.90227741643907382762501423525, −10.91335028499141244772470576035, −9.212565164174387439717095614639, −6.44771243960593321795590797896, −5.75333263909241173971323500780, −4.15686477265978088320503087188,
4.15686477265978088320503087188, 5.75333263909241173971323500780, 6.44771243960593321795590797896, 9.212565164174387439717095614639, 10.91335028499141244772470576035, 11.90227741643907382762501423525, 12.99212700705068834617634779508, 13.94588673316197709173832397868, 15.47778975023287695392669568729, 16.54043015310824634377959576393
