L(s) = 1 | + 2.30i·2-s + 3-s − 3.30·4-s + i·5-s + 2.30i·6-s − i·7-s − 3.00i·8-s − 2·9-s − 2.30·10-s + 1.60i·11-s − 3.30·12-s + 2.30·14-s + i·15-s + 0.302·16-s − 7.60·17-s − 4.60i·18-s + ⋯ |
L(s) = 1 | + 1.62i·2-s + 0.577·3-s − 1.65·4-s + 0.447i·5-s + 0.940i·6-s − 0.377i·7-s − 1.06i·8-s − 0.666·9-s − 0.728·10-s + 0.484i·11-s − 0.953·12-s + 0.615·14-s + 0.258i·15-s + 0.0756·16-s − 1.84·17-s − 1.08i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.449947 - 0.840735i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.449947 - 0.840735i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 - 1.60iT - 11T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 19 | \( 1 - 5.60iT - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + 6.21T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 3.60iT - 37T^{2} \) |
| 41 | \( 1 + 3iT - 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 9.21iT - 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 + 10.8iT - 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 4.81iT - 71T^{2} \) |
| 73 | \( 1 + 0.788iT - 73T^{2} \) |
| 79 | \( 1 - 5.21T + 79T^{2} \) |
| 83 | \( 1 - 9.21iT - 83T^{2} \) |
| 89 | \( 1 + 6.21iT - 89T^{2} \) |
| 97 | \( 1 - 8.39iT - 97T^{2} \) |
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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
−10.64517743373942906777073806399, −9.400906749326587168323894969927, −8.879607360961337695517342906078, −7.985501698632387911045098677466, −7.35463178769963724688243357952, −6.54615210084276495618198769249, −5.78237735914990167729217261077, −4.69568588763049536623614362532, −3.70913986405087788175847755240, −2.27669434407411454835486951992,
0.40549553316496237862697245364, 2.12010011695616190053804502294, 2.73929089197112560829296018160, 3.85044484808374319900907784407, 4.75462508972006341696082389192, 5.88406423917973855382087823922, 7.22906494741001589868184350980, 8.602386686639780786969910674099, 8.978337124948859797220329980388, 9.478373121470212273541852611688