| L(s) = 1 | − 3-s − 2·4-s + 2.14·5-s + 2.72·7-s + 9-s − 0.451·11-s + 2·12-s − 5.41·13-s − 2.14·15-s + 4·16-s − 3.24·17-s − 3.89·19-s − 4.28·20-s − 2.72·21-s + 23-s − 0.417·25-s − 27-s − 5.44·28-s + 29-s − 1.55·31-s + 0.451·33-s + 5.83·35-s − 2·36-s − 4.21·37-s + 5.41·39-s + 3.10·41-s + 10.6·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.957·5-s + 1.02·7-s + 0.333·9-s − 0.136·11-s + 0.577·12-s − 1.50·13-s − 0.552·15-s + 16-s − 0.785·17-s − 0.893·19-s − 0.957·20-s − 0.594·21-s + 0.208·23-s − 0.0834·25-s − 0.192·27-s − 1.02·28-s + 0.185·29-s − 0.279·31-s + 0.0785·33-s + 0.985·35-s − 0.333·36-s − 0.693·37-s + 0.866·39-s + 0.485·41-s + 1.62·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 - 2.14T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 + 0.451T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 4.21T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 + 0.834T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 6.69T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 + 1.82T + 71T^{2} \) |
| 73 | \( 1 + 8.72T + 73T^{2} \) |
| 79 | \( 1 + 8.73T + 79T^{2} \) |
| 83 | \( 1 - 0.403T + 83T^{2} \) |
| 89 | \( 1 - 4.22T + 89T^{2} \) |
| 97 | \( 1 + 1.86T + 97T^{2} \) |
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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
−8.950853692914178751141839720913, −8.012692487181896525800805424327, −7.25782865719920642421044431257, −6.18783880350779414961119038629, −5.41012875867174154304983869559, −4.77632996926550112704234403570, −4.20693104748393790998713950122, −2.55556271823300450854008985415, −1.56826377420068349399099209302, 0,
1.56826377420068349399099209302, 2.55556271823300450854008985415, 4.20693104748393790998713950122, 4.77632996926550112704234403570, 5.41012875867174154304983869559, 6.18783880350779414961119038629, 7.25782865719920642421044431257, 8.012692487181896525800805424327, 8.950853692914178751141839720913
