L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s − 2·9-s + 3·11-s + 12-s + 2·13-s + 4·14-s + 16-s − 6·17-s + 2·18-s − 4·21-s − 3·22-s − 6·23-s − 24-s − 5·25-s − 2·26-s − 5·27-s − 4·28-s + 2·31-s − 32-s + 3·33-s + 6·34-s − 2·36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s − 0.872·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s − 25-s − 0.392·26-s − 0.962·27-s − 0.755·28-s + 0.359·31-s − 0.176·32-s + 0.522·33-s + 1.02·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.745585523840483298014993195280, −9.059439427611583771768721665910, −8.575518973078556652839385214297, −7.44963857820717244434006439010, −6.41308622879988500397496204370, −5.98125887230847524844766016355, −4.05897099855705215218567080478, −3.20273446319884277273073329630, −2.04455641107678214923968718341, 0,
2.04455641107678214923968718341, 3.20273446319884277273073329630, 4.05897099855705215218567080478, 5.98125887230847524844766016355, 6.41308622879988500397496204370, 7.44963857820717244434006439010, 8.575518973078556652839385214297, 9.059439427611583771768721665910, 9.745585523840483298014993195280