L(s) = 1 | − 3-s + 9-s + 11-s − 2·13-s − 6·17-s + 2·19-s − 4·23-s − 27-s − 2·29-s − 33-s + 2·39-s − 2·41-s + 8·43-s + 6·51-s − 10·53-s − 2·57-s + 6·59-s + 8·61-s + 2·67-s + 4·69-s − 4·71-s + 6·73-s + 12·79-s + 81-s + 16·83-s + 2·87-s − 16·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s − 0.834·23-s − 0.192·27-s − 0.371·29-s − 0.174·33-s + 0.320·39-s − 0.312·41-s + 1.21·43-s + 0.840·51-s − 1.37·53-s − 0.264·57-s + 0.781·59-s + 1.02·61-s + 0.244·67-s + 0.481·69-s − 0.474·71-s + 0.702·73-s + 1.35·79-s + 1/9·81-s + 1.75·83-s + 0.214·87-s − 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.73604453421387, −12.36367468512704, −11.92702245723235, −11.49358804587364, −10.95892225272241, −10.78333457650026, −10.10909093569337, −9.551513467922364, −9.382547790138851, −8.769330674197258, −8.145777494600116, −7.819106723675423, −7.138650166415115, −6.779039022144011, −6.368319150104564, −5.804734549992026, −5.305851260387020, −4.840050216471652, −4.258938717168978, −3.914379258422462, −3.250012001619490, −2.456911406522606, −2.109845037752970, −1.411555785931904, −0.6417943767157702, 0,
0.6417943767157702, 1.411555785931904, 2.109845037752970, 2.456911406522606, 3.250012001619490, 3.914379258422462, 4.258938717168978, 4.840050216471652, 5.305851260387020, 5.804734549992026, 6.368319150104564, 6.779039022144011, 7.138650166415115, 7.819106723675423, 8.145777494600116, 8.769330674197258, 9.382547790138851, 9.551513467922364, 10.10909093569337, 10.78333457650026, 10.95892225272241, 11.49358804587364, 11.92702245723235, 12.36367468512704, 12.73604453421387