| L(s) = 1 | − 1.82·2-s + 1.31·4-s − 1.32·5-s + 2.50·7-s + 1.24·8-s + 2.41·10-s − 3.74·11-s + 2.41·13-s − 4.56·14-s − 4.90·16-s + 7.60·17-s − 7.85·19-s − 1.74·20-s + 6.81·22-s − 1.04·23-s − 3.23·25-s − 4.39·26-s + 3.29·28-s + 4.23·29-s + 8.90·31-s + 6.42·32-s − 13.8·34-s − 3.32·35-s + 8.05·37-s + 14.2·38-s − 1.65·40-s + 2.45·41-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.656·4-s − 0.593·5-s + 0.947·7-s + 0.441·8-s + 0.763·10-s − 1.12·11-s + 0.670·13-s − 1.21·14-s − 1.22·16-s + 1.84·17-s − 1.80·19-s − 0.389·20-s + 1.45·22-s − 0.217·23-s − 0.647·25-s − 0.862·26-s + 0.622·28-s + 0.787·29-s + 1.59·31-s + 1.13·32-s − 2.37·34-s − 0.562·35-s + 1.32·37-s + 2.31·38-s − 0.262·40-s + 0.382·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6965745835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6965745835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 - 2.50T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 8.90T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 - 2.45T + 41T^{2} \) |
| 43 | \( 1 - 0.929T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 - 9.49T + 59T^{2} \) |
| 61 | \( 1 - 1.53T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 8.81T + 97T^{2} \) |
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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
−10.41411446138062245572882484762, −9.838190563890401508332075821464, −8.482618472192929407323212974333, −8.124065611978952258481059810690, −7.62897539014139354784118434227, −6.29000688594150718399179508051, −5.02295002839039582220323889348, −4.02306741051576069263411560096, −2.35802096495412884132953237705, −0.896330266672838818741229710354,
0.896330266672838818741229710354, 2.35802096495412884132953237705, 4.02306741051576069263411560096, 5.02295002839039582220323889348, 6.29000688594150718399179508051, 7.62897539014139354784118434227, 8.124065611978952258481059810690, 8.482618472192929407323212974333, 9.838190563890401508332075821464, 10.41411446138062245572882484762
