| L(s) = 1 | + 2-s + 4-s + 8-s − 3·9-s − 2·11-s + 2·13-s + 16-s − 2·17-s − 3·18-s − 4·19-s − 2·22-s − 23-s + 2·26-s + 6·29-s − 2·31-s + 32-s − 2·34-s − 3·36-s − 4·37-s − 4·38-s + 12·41-s − 6·43-s − 2·44-s − 46-s + 6·47-s + 2·52-s + 12·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.426·22-s − 0.208·23-s + 0.392·26-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 0.657·37-s − 0.648·38-s + 1.87·41-s − 0.914·43-s − 0.301·44-s − 0.147·46-s + 0.875·47-s + 0.277·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.65060631635100, −13.97881329534598, −13.71641910934768, −13.18349330198656, −12.63827633114610, −12.15408138861924, −11.63490308911354, −11.11872543839433, −10.52526641636605, −10.39813394289717, −9.423610132979513, −8.740988573046198, −8.531768408682384, −7.807474216309793, −7.266295788698300, −6.547373272256775, −6.070382581866028, −5.649305802131999, −4.990654346556279, −4.396753579590640, −3.825213417037990, −3.109811598431891, −2.522900701334638, −2.027020839973339, −0.9603923316295676, 0,
0.9603923316295676, 2.027020839973339, 2.522900701334638, 3.109811598431891, 3.825213417037990, 4.396753579590640, 4.990654346556279, 5.649305802131999, 6.070382581866028, 6.547373272256775, 7.266295788698300, 7.807474216309793, 8.531768408682384, 8.740988573046198, 9.423610132979513, 10.39813394289717, 10.52526641636605, 11.11872543839433, 11.63490308911354, 12.15408138861924, 12.63827633114610, 13.18349330198656, 13.71641910934768, 13.97881329534598, 14.65060631635100
