| L(s) = 1 | − 2·11-s − 13-s + 2·17-s − 8·19-s + 4·23-s − 6·29-s − 4·31-s + 6·37-s + 12·41-s + 4·43-s − 6·47-s − 7·49-s + 2·53-s − 14·59-s − 10·61-s − 4·67-s − 2·71-s + 2·73-s − 8·79-s − 14·83-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.277·13-s + 0.485·17-s − 1.83·19-s + 0.834·23-s − 1.11·29-s − 0.718·31-s + 0.986·37-s + 1.87·41-s + 0.609·43-s − 0.875·47-s − 49-s + 0.274·53-s − 1.82·59-s − 1.28·61-s − 0.488·67-s − 0.237·71-s + 0.234·73-s − 0.900·79-s − 1.53·83-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6931057308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6931057308\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.01191679084673, −12.78221146714963, −12.34283241942062, −11.67164459205524, −11.09095019376256, −10.80266099292762, −10.47812011369802, −9.728653563940932, −9.277707452717561, −9.006029218401364, −8.226133657667711, −7.884483206354616, −7.395425438324472, −6.938538543849972, −6.145609019984790, −5.946377683431642, −5.330416262376182, −4.584268592288507, −4.381069307002156, −3.652804844768969, −2.956203929531061, −2.545260619944360, −1.843981791738232, −1.245583075417232, −0.2354035728911013,
0.2354035728911013, 1.245583075417232, 1.843981791738232, 2.545260619944360, 2.956203929531061, 3.652804844768969, 4.381069307002156, 4.584268592288507, 5.330416262376182, 5.946377683431642, 6.145609019984790, 6.938538543849972, 7.395425438324472, 7.884483206354616, 8.226133657667711, 9.006029218401364, 9.277707452717561, 9.728653563940932, 10.47812011369802, 10.80266099292762, 11.09095019376256, 11.67164459205524, 12.34283241942062, 12.78221146714963, 13.01191679084673
