| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 13-s − 15-s + 16-s + 6·17-s − 18-s − 4·19-s + 20-s + 8·23-s + 24-s + 25-s + 26-s − 27-s − 6·29-s + 30-s − 8·31-s − 32-s − 6·34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.91533050124373, −14.48058830040576, −13.88308516756261, −13.05769240429126, −12.72148920243124, −12.33555095435778, −11.63442369135002, −11.00560722974052, −10.78895121129253, −10.16795440085088, −9.632483322156037, −9.127048245665031, −8.735527794545028, −7.775087248898594, −7.577500531038863, −6.789003918714998, −6.432750343044056, −5.642005268456999, −5.239576664384561, −4.711095721274938, −3.560712434406409, −3.302307126763878, −2.203551916873515, −1.684637861469639, −0.9013762334379479, 0,
0.9013762334379479, 1.684637861469639, 2.203551916873515, 3.302307126763878, 3.560712434406409, 4.711095721274938, 5.239576664384561, 5.642005268456999, 6.432750343044056, 6.789003918714998, 7.577500531038863, 7.775087248898594, 8.735527794545028, 9.127048245665031, 9.632483322156037, 10.16795440085088, 10.78895121129253, 11.00560722974052, 11.63442369135002, 12.33555095435778, 12.72148920243124, 13.05769240429126, 13.88308516756261, 14.48058830040576, 14.91533050124373
