| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 3·11-s − 12-s − 13-s − 14-s + 15-s + 16-s + 6·17-s + 18-s − 7·19-s − 20-s + 21-s − 3·22-s + 3·23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.904·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.60·19-s − 0.223·20-s + 0.218·21-s − 0.639·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−12.72233879532248, −12.31784746841090, −11.89696553788556, −11.41890124647868, −10.97693664068886, −10.52508857129967, −10.20248094243235, −9.667604868763415, −9.217811970252173, −8.337211079843214, −8.130438823499794, −7.669677752196247, −6.969662609662926, −6.734527364415229, −6.240576171781856, −5.580189635767844, −5.147756550842770, −4.986475114683387, −4.208187622735312, −3.659013673555238, −3.363194068127818, −2.677586600617577, −2.086491430158435, −1.500158843212138, −0.6206761792211673, 0,
0.6206761792211673, 1.500158843212138, 2.086491430158435, 2.677586600617577, 3.363194068127818, 3.659013673555238, 4.208187622735312, 4.986475114683387, 5.147756550842770, 5.580189635767844, 6.240576171781856, 6.734527364415229, 6.969662609662926, 7.669677752196247, 8.130438823499794, 8.337211079843214, 9.217811970252173, 9.667604868763415, 10.20248094243235, 10.52508857129967, 10.97693664068886, 11.41890124647868, 11.89696553788556, 12.31784746841090, 12.72233879532248
