L(s) = 1 | + 2-s − 2·3-s − 4-s + 5-s − 2·6-s + 4·7-s − 3·8-s + 9-s + 10-s − 2·11-s + 2·12-s + 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 8·21-s − 2·22-s − 6·23-s + 6·24-s + 25-s + 4·27-s − 4·28-s + 2·29-s − 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.577·12-s + 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 1.74·21-s − 0.426·22-s − 1.25·23-s + 1.22·24-s + 1/5·25-s + 0.769·27-s − 0.755·28-s + 0.371·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.492934938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.492934938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.34630410671677379936192033935, −9.532805961208409547451766916679, −8.353551523811345329615757792465, −7.69003698049592494171610769456, −6.24360515405325801186702466344, −5.56389466213996946942866194317, −5.01117336241618809550332242437, −4.27130437726285993901225335150, −2.71745145781960607840515064653, −0.996154674950302563103745756297,
0.996154674950302563103745756297, 2.71745145781960607840515064653, 4.27130437726285993901225335150, 5.01117336241618809550332242437, 5.56389466213996946942866194317, 6.24360515405325801186702466344, 7.69003698049592494171610769456, 8.353551523811345329615757792465, 9.532805961208409547451766916679, 10.34630410671677379936192033935