| L(s) = 1 | − 3·5-s + 7-s − 13-s + 5·17-s + 4·19-s + 4·23-s + 4·25-s − 7·29-s − 4·31-s − 3·35-s − 5·37-s + 9·41-s + 49-s − 3·53-s − 2·61-s + 3·65-s + 8·67-s − 6·73-s − 15·85-s + 13·89-s − 91-s − 12·95-s − 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 0.277·13-s + 1.21·17-s + 0.917·19-s + 0.834·23-s + 4/5·25-s − 1.29·29-s − 0.718·31-s − 0.507·35-s − 0.821·37-s + 1.40·41-s + 1/7·49-s − 0.412·53-s − 0.256·61-s + 0.372·65-s + 0.977·67-s − 0.702·73-s − 1.62·85-s + 1.37·89-s − 0.104·91-s − 1.23·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−14.64781814184186, −14.18998036172329, −13.56198198393469, −12.88143571488664, −12.46081321975824, −11.97612629378204, −11.55429320507054, −11.03885334283115, −10.69023053004304, −9.904101284476199, −9.355028422900854, −8.932728476598125, −8.131546440424899, −7.795609467483752, −7.330947087434467, −7.004567600178700, −6.040061854885849, −5.391688304674241, −5.041709469688070, −4.262324447939460, −3.680133507665726, −3.295107017738471, −2.545318016256335, −1.572536337010777, −0.8843169926989916, 0,
0.8843169926989916, 1.572536337010777, 2.545318016256335, 3.295107017738471, 3.680133507665726, 4.262324447939460, 5.041709469688070, 5.391688304674241, 6.040061854885849, 7.004567600178700, 7.330947087434467, 7.795609467483752, 8.131546440424899, 8.932728476598125, 9.355028422900854, 9.904101284476199, 10.69023053004304, 11.03885334283115, 11.55429320507054, 11.97612629378204, 12.46081321975824, 12.88143571488664, 13.56198198393469, 14.18998036172329, 14.64781814184186
