| L(s) = 1 | + 5-s − 1.73·7-s − 1.73·11-s + 3·13-s + 4·17-s − 6.92·19-s − 8.66·23-s − 4·25-s − 29-s + 5.19·31-s − 1.73·35-s + 8·37-s + 5·41-s + 8.66·43-s + 12.1·47-s − 4·49-s + 8·53-s − 1.73·55-s − 1.73·59-s + 7·61-s + 3·65-s + 8.66·67-s + 3.46·71-s − 12·73-s + 2.99·77-s − 5.19·79-s + 8.66·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.654·7-s − 0.522·11-s + 0.832·13-s + 0.970·17-s − 1.58·19-s − 1.80·23-s − 0.800·25-s − 0.185·29-s + 0.933·31-s − 0.292·35-s + 1.31·37-s + 0.780·41-s + 1.32·43-s + 1.76·47-s − 0.571·49-s + 1.09·53-s − 0.233·55-s − 0.225·59-s + 0.896·61-s + 0.372·65-s + 1.05·67-s + 0.411·71-s − 1.40·73-s + 0.341·77-s − 0.584·79-s + 0.950·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737020376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737020376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 1.73T + 7T^{2} \) |
| 11 | \( 1 + 1.73T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 8.66T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 8.66T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−8.116171855751397384179746808633, −7.66717581180779453094975580712, −6.58385023544714383895824120120, −5.92923841712819960777507591473, −5.68266318561014168260574872829, −4.28099758158078081782584129692, −3.87888384228903888205385556982, −2.71064986024512970671808825689, −2.03616995346871017489508460117, −0.69977005142971179974784360563,
0.69977005142971179974784360563, 2.03616995346871017489508460117, 2.71064986024512970671808825689, 3.87888384228903888205385556982, 4.28099758158078081782584129692, 5.68266318561014168260574872829, 5.92923841712819960777507591473, 6.58385023544714383895824120120, 7.66717581180779453094975580712, 8.116171855751397384179746808633
