| L(s) = 1 | − 0.555·2-s − 3-s − 1.69·4-s − 1.46·5-s + 0.555·6-s + 7-s + 2.05·8-s + 9-s + 0.812·10-s + 4.65·11-s + 1.69·12-s − 1.76·13-s − 0.555·14-s + 1.46·15-s + 2.24·16-s + 0.856·17-s − 0.555·18-s + 1.84·19-s + 2.47·20-s − 21-s − 2.58·22-s − 5.80·23-s − 2.05·24-s − 2.86·25-s + 0.982·26-s − 27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.577·3-s − 0.845·4-s − 0.654·5-s + 0.226·6-s + 0.377·7-s + 0.725·8-s + 0.333·9-s + 0.256·10-s + 1.40·11-s + 0.488·12-s − 0.490·13-s − 0.148·14-s + 0.377·15-s + 0.560·16-s + 0.207·17-s − 0.130·18-s + 0.422·19-s + 0.553·20-s − 0.218·21-s − 0.550·22-s − 1.21·23-s − 0.418·24-s − 0.572·25-s + 0.192·26-s − 0.192·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 0.555T + 2T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 11 | \( 1 - 4.65T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 0.856T + 17T^{2} \) |
| 19 | \( 1 - 1.84T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 3.84T + 29T^{2} \) |
| 31 | \( 1 - 0.134T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 7.45T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 0.949T + 47T^{2} \) |
| 53 | \( 1 - 0.392T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.57T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 2.90T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.45613641172938287173716315601, −7.00198116768176924573350577148, −6.03401077309533197091091710103, −5.37326579160973420667310469468, −4.52149510030098986822508489582, −4.08786628571746055053350673363, −3.38298745159122399564197742123, −1.88778200197293384402329677371, −1.02722734448099707595524439949, 0,
1.02722734448099707595524439949, 1.88778200197293384402329677371, 3.38298745159122399564197742123, 4.08786628571746055053350673363, 4.52149510030098986822508489582, 5.37326579160973420667310469468, 6.03401077309533197091091710103, 7.00198116768176924573350577148, 7.45613641172938287173716315601
