| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 2·11-s + 6·13-s + 14-s + 16-s − 4·17-s − 6·19-s + 2·22-s + 8·23-s − 6·26-s − 28-s − 6·29-s − 2·31-s − 32-s + 4·34-s + 4·37-s + 6·38-s − 2·41-s + 4·43-s − 2·44-s − 8·46-s − 8·47-s + 49-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.426·22-s + 1.66·23-s − 1.17·26-s − 0.188·28-s − 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.685·34-s + 0.657·37-s + 0.973·38-s − 0.312·41-s + 0.609·43-s − 0.301·44-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.498382546757203020239258495433, −7.68014441884016908563546149791, −6.74997205639578676060723695878, −6.29297174383752191175801963508, −5.40211382680862004862464014820, −4.31239215157580298649474530551, −3.41017601767058417655679485415, −2.45105548459720294833734763025, −1.37656617987862700655396503009, 0,
1.37656617987862700655396503009, 2.45105548459720294833734763025, 3.41017601767058417655679485415, 4.31239215157580298649474530551, 5.40211382680862004862464014820, 6.29297174383752191175801963508, 6.74997205639578676060723695878, 7.68014441884016908563546149791, 8.498382546757203020239258495433
