L(s) = 1 | − 2-s − 3-s + 4-s − 1.03·5-s + 6-s − 2.85·7-s − 8-s + 9-s + 1.03·10-s − 4.15·11-s − 12-s + 2.83·13-s + 2.85·14-s + 1.03·15-s + 16-s − 17-s − 18-s − 2.04·19-s − 1.03·20-s + 2.85·21-s + 4.15·22-s + 7.42·23-s + 24-s − 3.93·25-s − 2.83·26-s − 27-s − 2.85·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.461·5-s + 0.408·6-s − 1.07·7-s − 0.353·8-s + 0.333·9-s + 0.326·10-s − 1.25·11-s − 0.288·12-s + 0.787·13-s + 0.762·14-s + 0.266·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.469·19-s − 0.230·20-s + 0.622·21-s + 0.884·22-s + 1.54·23-s + 0.204·24-s − 0.787·25-s − 0.556·26-s − 0.192·27-s − 0.539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3292492373\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3292492373\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.83T + 13T^{2} \) |
| 19 | \( 1 + 2.04T + 19T^{2} \) |
| 23 | \( 1 - 7.42T + 23T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 4.78T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 0.900T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 0.396T + 53T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 9.87T + 79T^{2} \) |
| 83 | \( 1 + 4.49T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−8.051261887692231740432701001546, −7.38821675756413887520197930617, −6.72003657095986450018473373431, −6.08717755247145637893371940571, −5.38727284278733122973657024629, −4.47140866109096602796143455267, −3.45460592769075059758918390245, −2.83516164587858749466743708660, −1.63586034635095896749478367775, −0.33975553865699725850739058636,
0.33975553865699725850739058636, 1.63586034635095896749478367775, 2.83516164587858749466743708660, 3.45460592769075059758918390245, 4.47140866109096602796143455267, 5.38727284278733122973657024629, 6.08717755247145637893371940571, 6.72003657095986450018473373431, 7.38821675756413887520197930617, 8.051261887692231740432701001546