| L(s) = 1 | − 3·11-s − 2·13-s + 2·19-s − 7·23-s + 3·29-s − 6·31-s + 3·37-s + 5·43-s + 2·47-s + 2·53-s + 10·59-s + 8·61-s − 9·67-s + 9·71-s − 8·73-s + 79-s + 14·83-s − 6·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s − 0.554·13-s + 0.458·19-s − 1.45·23-s + 0.557·29-s − 1.07·31-s + 0.493·37-s + 0.762·43-s + 0.291·47-s + 0.274·53-s + 1.30·59-s + 1.02·61-s − 1.09·67-s + 1.06·71-s − 0.936·73-s + 0.112·79-s + 1.53·83-s − 0.635·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.45035430580785, −12.92375122949141, −12.48841407417818, −11.98135457802770, −11.65337118328309, −11.00312421797212, −10.48399606737181, −10.22548297239420, −9.523872436577417, −9.330720957705082, −8.540136382293583, −8.050627165727671, −7.750501289190834, −7.143025608905630, −6.726512126188462, −5.989783582688856, −5.523177573138689, −5.193348680672798, −4.447590325789175, −3.992563303855888, −3.403043083201838, −2.601673807076160, −2.342314968815968, −1.576952481647117, −0.7188780215056183, 0,
0.7188780215056183, 1.576952481647117, 2.342314968815968, 2.601673807076160, 3.403043083201838, 3.992563303855888, 4.447590325789175, 5.193348680672798, 5.523177573138689, 5.989783582688856, 6.726512126188462, 7.143025608905630, 7.750501289190834, 8.050627165727671, 8.540136382293583, 9.330720957705082, 9.523872436577417, 10.22548297239420, 10.48399606737181, 11.00312421797212, 11.65337118328309, 11.98135457802770, 12.48841407417818, 12.92375122949141, 13.45035430580785
