| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 13-s + 16-s − 20-s + 2·23-s + 25-s + 26-s − 8·29-s − 4·31-s − 32-s + 2·37-s + 40-s + 6·41-s + 4·43-s − 2·46-s − 4·47-s − 7·49-s − 50-s − 52-s + 2·53-s + 8·58-s − 12·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.277·13-s + 1/4·16-s − 0.223·20-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 1.48·29-s − 0.718·31-s − 0.176·32-s + 0.328·37-s + 0.158·40-s + 0.937·41-s + 0.609·43-s − 0.294·46-s − 0.583·47-s − 49-s − 0.141·50-s − 0.138·52-s + 0.274·53-s + 1.05·58-s − 1.53·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1264915787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1264915787\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.58223621778048, −12.09041626894311, −11.42638366036220, −11.27679010095697, −10.76109961391033, −10.39402695116812, −9.682306526629291, −9.287434889437039, −9.157004729166391, −8.345759615681351, −7.986834521344427, −7.633126675505545, −6.956352889272997, −6.870086996860092, −6.028508170372437, −5.593310620801535, −5.175975214604652, −4.334509308688978, −4.083197649270706, −3.340216812093686, −2.855313910857274, −2.314216924030672, −1.557565950637513, −1.153509899260552, −0.1085187174856325,
0.1085187174856325, 1.153509899260552, 1.557565950637513, 2.314216924030672, 2.855313910857274, 3.340216812093686, 4.083197649270706, 4.334509308688978, 5.175975214604652, 5.593310620801535, 6.028508170372437, 6.870086996860092, 6.956352889272997, 7.633126675505545, 7.986834521344427, 8.345759615681351, 9.157004729166391, 9.287434889437039, 9.682306526629291, 10.39402695116812, 10.76109961391033, 11.27679010095697, 11.42638366036220, 12.09041626894311, 12.58223621778048
