L(s) = 1 | + 5-s − 2·11-s − 2·13-s − 6·17-s − 2·19-s − 8·23-s + 25-s + 4·29-s − 2·31-s + 2·37-s + 6·41-s + 4·43-s − 4·47-s + 2·53-s − 2·55-s + 4·59-s + 12·61-s − 2·65-s − 16·67-s − 6·71-s + 14·73-s + 8·79-s − 4·83-s − 6·85-s + 10·89-s − 2·95-s + 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.554·13-s − 1.45·17-s − 0.458·19-s − 1.66·23-s + 1/5·25-s + 0.742·29-s − 0.359·31-s + 0.328·37-s + 0.937·41-s + 0.609·43-s − 0.583·47-s + 0.274·53-s − 0.269·55-s + 0.520·59-s + 1.53·61-s − 0.248·65-s − 1.95·67-s − 0.712·71-s + 1.63·73-s + 0.900·79-s − 0.439·83-s − 0.650·85-s + 1.05·89-s − 0.205·95-s + 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.56492152165195, −13.17197252475137, −12.78091832345263, −12.21376429045894, −11.79658839426079, −11.15394879478875, −10.77465496363774, −10.22022889175412, −9.877371752574828, −9.319260542548946, −8.760965423859880, −8.372977501568993, −7.729060430878239, −7.350016219914808, −6.626648651224607, −6.236994760977812, −5.776584648715589, −5.096682884543258, −4.593190432014393, −4.130989290235861, −3.489594351749284, −2.535659619842163, −2.346498013204382, −1.758112276709844, −0.7260861682941535, 0,
0.7260861682941535, 1.758112276709844, 2.346498013204382, 2.535659619842163, 3.489594351749284, 4.130989290235861, 4.593190432014393, 5.096682884543258, 5.776584648715589, 6.236994760977812, 6.626648651224607, 7.350016219914808, 7.729060430878239, 8.372977501568993, 8.760965423859880, 9.319260542548946, 9.877371752574828, 10.22022889175412, 10.77465496363774, 11.15394879478875, 11.79658839426079, 12.21376429045894, 12.78091832345263, 13.17197252475137, 13.56492152165195