L(s) = 1 | + 3-s + 0.808·5-s − 2.11·7-s + 9-s − 0.503·11-s − 3.72·13-s + 0.808·15-s + 2.72·17-s + 6.39·19-s − 2.11·21-s − 23-s − 4.34·25-s + 27-s − 29-s + 0.365·31-s − 0.503·33-s − 1.71·35-s + 2.18·37-s − 3.72·39-s − 8.11·41-s − 9.75·43-s + 0.808·45-s − 3.69·47-s − 2.50·49-s + 2.72·51-s + 7.96·53-s − 0.406·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.361·5-s − 0.800·7-s + 0.333·9-s − 0.151·11-s − 1.03·13-s + 0.208·15-s + 0.661·17-s + 1.46·19-s − 0.462·21-s − 0.208·23-s − 0.869·25-s + 0.192·27-s − 0.185·29-s + 0.0657·31-s − 0.0875·33-s − 0.289·35-s + 0.358·37-s − 0.596·39-s − 1.26·41-s − 1.48·43-s + 0.120·45-s − 0.538·47-s − 0.358·49-s + 0.382·51-s + 1.09·53-s − 0.0548·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - 0.808T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 11 | \( 1 + 0.503T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 6.39T + 19T^{2} \) |
| 31 | \( 1 - 0.365T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 + 9.75T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 + 4.14T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 0.758T + 83T^{2} \) |
| 89 | \( 1 - 8.63T + 89T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−7.56133702591562874401707581973, −6.88788872229737253225543893381, −6.14305075525115472218573085619, −5.35163335202575961867062609359, −4.76730953320313926749054714683, −3.61252320910209043212148052171, −3.17000361078665317640511774529, −2.32948189204856278103360482341, −1.39186691283582504464706963573, 0,
1.39186691283582504464706963573, 2.32948189204856278103360482341, 3.17000361078665317640511774529, 3.61252320910209043212148052171, 4.76730953320313926749054714683, 5.35163335202575961867062609359, 6.14305075525115472218573085619, 6.88788872229737253225543893381, 7.56133702591562874401707581973