L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s + 2·14-s + 15-s + 16-s − 18-s + 19-s − 20-s + 2·21-s + 24-s + 25-s − 2·26-s − 27-s − 2·28-s + 6·29-s − 30-s − 2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.182·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7489127060\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7489127060\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.00433250355279, −12.74025616161256, −12.21913023212331, −11.71352911361300, −11.39096256487496, −10.80319295230377, −10.31740587545032, −10.10318510403779, −9.336591893945004, −9.009040287724286, −8.504116236633916, −7.926128967516555, −7.359051760635719, −7.046603270302370, −6.310983695573444, −6.127220826484295, −5.483760743941311, −4.816769400085545, −4.228934404126762, −3.637217647861480, −3.058242455224540, −2.529132026536333, −1.629083320640718, −1.026398374056038, −0.3446372550228635,
0.3446372550228635, 1.026398374056038, 1.629083320640718, 2.529132026536333, 3.058242455224540, 3.637217647861480, 4.228934404126762, 4.816769400085545, 5.483760743941311, 6.127220826484295, 6.310983695573444, 7.046603270302370, 7.359051760635719, 7.926128967516555, 8.504116236633916, 9.009040287724286, 9.336591893945004, 10.10318510403779, 10.31740587545032, 10.80319295230377, 11.39096256487496, 11.71352911361300, 12.21913023212331, 12.74025616161256, 13.00433250355279