| L(s) = 1 | − 3-s + 9-s − 2·11-s + 2·13-s − 6·19-s − 27-s − 6·29-s + 10·31-s + 2·33-s − 2·39-s − 6·41-s + 8·43-s − 12·47-s − 6·53-s + 6·57-s + 6·61-s − 4·67-s − 6·71-s + 14·73-s − 4·79-s + 81-s + 6·87-s + 6·89-s − 10·93-s + 2·97-s − 2·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.37·19-s − 0.192·27-s − 1.11·29-s + 1.79·31-s + 0.348·33-s − 0.320·39-s − 0.937·41-s + 1.21·43-s − 1.75·47-s − 0.824·53-s + 0.794·57-s + 0.768·61-s − 0.488·67-s − 0.712·71-s + 1.63·73-s − 0.450·79-s + 1/9·81-s + 0.643·87-s + 0.635·89-s − 1.03·93-s + 0.203·97-s − 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−14.59267809682255, −14.09984035360907, −13.40458249144442, −13.00869316161395, −12.71198110357043, −11.96380367231049, −11.54833233774449, −10.97424175564229, −10.61295485004718, −10.04459890085869, −9.590533052836109, −8.840317264356716, −8.331819306851324, −7.905859810597850, −7.217706376696592, −6.579304752908564, −6.151841879545849, −5.688937162030366, −4.824836474530673, −4.599054384449614, −3.766779823456666, −3.186516344279352, −2.321561983929457, −1.750025784371183, −0.8187339407647973, 0,
0.8187339407647973, 1.750025784371183, 2.321561983929457, 3.186516344279352, 3.766779823456666, 4.599054384449614, 4.824836474530673, 5.688937162030366, 6.151841879545849, 6.579304752908564, 7.217706376696592, 7.905859810597850, 8.331819306851324, 8.840317264356716, 9.590533052836109, 10.04459890085869, 10.61295485004718, 10.97424175564229, 11.54833233774449, 11.96380367231049, 12.71198110357043, 13.00869316161395, 13.40458249144442, 14.09984035360907, 14.59267809682255
