L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 4·11-s + 12-s + 6·13-s + 2·14-s + 16-s − 4·17-s + 18-s + 19-s + 2·21-s + 4·22-s − 4·23-s + 24-s + 6·26-s + 27-s + 2·28-s + 6·29-s − 6·31-s + 32-s + 4·33-s − 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s + 0.436·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 1.17·26-s + 0.192·27-s + 0.377·28-s + 1.11·29-s − 1.07·31-s + 0.176·32-s + 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.471255152\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.471255152\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.493317052900361757548402427478, −8.325739778927020131341614043820, −7.04027689541733850595684241447, −6.56633307744407450846153899611, −5.69352809902901260248738404390, −4.72637332509390000724936007504, −3.91294025037034861535747385811, −3.42112266621549540078821297796, −2.05767314382937436554137954042, −1.34299165269447374241539013096,
1.34299165269447374241539013096, 2.05767314382937436554137954042, 3.42112266621549540078821297796, 3.91294025037034861535747385811, 4.72637332509390000724936007504, 5.69352809902901260248738404390, 6.56633307744407450846153899611, 7.04027689541733850595684241447, 8.325739778927020131341614043820, 8.493317052900361757548402427478