| L(s) = 1 | − 2-s − 3-s + 4-s + 3·5-s + 6-s − 7-s − 8-s + 9-s − 3·10-s + 4·11-s − 12-s + 3·13-s + 14-s − 3·15-s + 16-s − 4·17-s − 18-s + 3·20-s + 21-s − 4·22-s + 23-s + 24-s + 4·25-s − 3·26-s − 27-s − 28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.20·11-s − 0.288·12-s + 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.670·20-s + 0.218·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s − 0.192·27-s − 0.188·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.96666080894727, −12.12589473795351, −11.72492149982124, −11.45709679145684, −10.85285604430687, −10.49350242887421, −9.953774209664593, −9.620484946036125, −9.235412044308625, −8.813242561587085, −8.348553745537489, −7.787749045660196, −6.975987556655940, −6.675198921908798, −6.278700701544139, −6.080061471178610, −5.406753615533549, −4.915112970506403, −4.207540201950063, −3.761062424078827, −3.024592563587354, −2.459319917309407, −1.795790914230662, −1.400941192450197, −0.8592982037842858, 0,
0.8592982037842858, 1.400941192450197, 1.795790914230662, 2.459319917309407, 3.024592563587354, 3.761062424078827, 4.207540201950063, 4.915112970506403, 5.406753615533549, 6.080061471178610, 6.278700701544139, 6.675198921908798, 6.975987556655940, 7.787749045660196, 8.348553745537489, 8.813242561587085, 9.235412044308625, 9.620484946036125, 9.953774209664593, 10.49350242887421, 10.85285604430687, 11.45709679145684, 11.72492149982124, 12.12589473795351, 12.96666080894727
