L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s − 11-s + 12-s + 2·13-s + 14-s + 16-s + 18-s + 21-s − 22-s − 23-s + 24-s + 2·26-s + 27-s + 28-s + 6·29-s + 8·31-s + 32-s − 33-s + 36-s + 3·37-s + 2·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s − 0.213·22-s − 0.208·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.493·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.473009536\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.473009536\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.83027572980773, −12.27082564286500, −11.84172193107098, −11.36527618379861, −10.87433343290750, −10.49889161257970, −9.934497785253681, −9.529454475539621, −8.940258674020989, −8.326610394142494, −8.107755306436050, −7.627781059369404, −7.035237893489436, −6.509534288493191, −6.104924026614043, −5.576321510873842, −4.954281690679555, −4.482394579806650, −4.125576093416728, −3.480079529354469, −2.952693051382855, −2.438641079122149, −2.002695495646940, −1.121748148694633, −0.7208327204058627,
0.7208327204058627, 1.121748148694633, 2.002695495646940, 2.438641079122149, 2.952693051382855, 3.480079529354469, 4.125576093416728, 4.482394579806650, 4.954281690679555, 5.576321510873842, 6.104924026614043, 6.509534288493191, 7.035237893489436, 7.627781059369404, 8.107755306436050, 8.326610394142494, 8.940258674020989, 9.529454475539621, 9.934497785253681, 10.49889161257970, 10.87433343290750, 11.36527618379861, 11.84172193107098, 12.27082564286500, 12.83027572980773