| L(s) = 1 | − 0.845·2-s − 3-s − 1.28·4-s + 0.712·5-s + 0.845·6-s + 7-s + 2.77·8-s + 9-s − 0.602·10-s + 4.61·11-s + 1.28·12-s − 2.19·13-s − 0.845·14-s − 0.712·15-s + 0.220·16-s − 0.845·18-s − 3.21·19-s − 0.914·20-s − 21-s − 3.90·22-s + 6.83·23-s − 2.77·24-s − 4.49·25-s + 1.85·26-s − 27-s − 1.28·28-s − 1.30·29-s + ⋯ |
| L(s) = 1 | − 0.598·2-s − 0.577·3-s − 0.642·4-s + 0.318·5-s + 0.345·6-s + 0.377·7-s + 0.982·8-s + 0.333·9-s − 0.190·10-s + 1.39·11-s + 0.370·12-s − 0.609·13-s − 0.226·14-s − 0.183·15-s + 0.0550·16-s − 0.199·18-s − 0.738·19-s − 0.204·20-s − 0.218·21-s − 0.832·22-s + 1.42·23-s − 0.567·24-s − 0.898·25-s + 0.364·26-s − 0.192·27-s − 0.242·28-s − 0.242·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.845T + 2T^{2} \) |
| 5 | \( 1 - 0.712T + 5T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 19 | \( 1 + 3.21T + 19T^{2} \) |
| 23 | \( 1 - 6.83T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 - 3.29T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 6.15T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 + 7.88T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 9.88T + 97T^{2} \) |
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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
−7.71768414187715470867026638309, −7.04800967312933209363902325478, −6.39925872958073427970855676257, −5.52277043939273241183801361922, −4.78745887961738918717926609746, −4.26204883509625389606957284718, −3.32727377179594895358482666826, −1.88918095103163547700151800563, −1.21741262779211422368335122093, 0,
1.21741262779211422368335122093, 1.88918095103163547700151800563, 3.32727377179594895358482666826, 4.26204883509625389606957284718, 4.78745887961738918717926609746, 5.52277043939273241183801361922, 6.39925872958073427970855676257, 7.04800967312933209363902325478, 7.71768414187715470867026638309
