| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 2·11-s − 6·13-s + 4·14-s + 16-s + 4·17-s + 2·19-s − 20-s + 2·22-s + 4·23-s + 25-s − 6·26-s + 4·28-s − 2·29-s + 8·31-s + 32-s + 4·34-s − 4·35-s − 4·37-s + 2·38-s − 40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s + 0.755·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.685·34-s − 0.676·35-s − 0.657·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.375709740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.375709740\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−13.36645315955312, −12.49074227713214, −12.27982261583374, −11.87643501247879, −11.54910011673782, −10.97159876005830, −10.59716604721420, −9.793521373296762, −9.663575987793938, −8.792975131145035, −8.254722451175760, −7.908382718297727, −7.341985595618616, −7.003744072621215, −6.489373466628992, −5.466913917716611, −5.346926942938031, −4.818305559264210, −4.326231759564160, −3.858636753978153, −3.011941240655837, −2.694160425825780, −1.828171518791007, −1.355388540605258, −0.6058547787272563,
0.6058547787272563, 1.355388540605258, 1.828171518791007, 2.694160425825780, 3.011941240655837, 3.858636753978153, 4.326231759564160, 4.818305559264210, 5.346926942938031, 5.466913917716611, 6.489373466628992, 7.003744072621215, 7.341985595618616, 7.908382718297727, 8.254722451175760, 8.792975131145035, 9.663575987793938, 9.793521373296762, 10.59716604721420, 10.97159876005830, 11.54910011673782, 11.87643501247879, 12.27982261583374, 12.49074227713214, 13.36645315955312
