| L(s) = 1 | − 5-s + 7-s + 2·11-s − 6·17-s − 6·19-s + 8·23-s + 25-s + 2·29-s − 10·31-s − 35-s − 2·37-s + 10·41-s − 4·43-s − 8·47-s + 49-s − 4·53-s − 2·55-s − 8·59-s + 6·61-s − 12·67-s − 6·71-s + 12·73-s + 2·77-s − 8·79-s − 4·83-s + 6·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.603·11-s − 1.45·17-s − 1.37·19-s + 1.66·23-s + 1/5·25-s + 0.371·29-s − 1.79·31-s − 0.169·35-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.549·53-s − 0.269·55-s − 1.04·59-s + 0.768·61-s − 1.46·67-s − 0.712·71-s + 1.40·73-s + 0.227·77-s − 0.900·79-s − 0.439·83-s + 0.650·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5893079388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5893079388\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.04138958809108, −12.55193478011075, −12.15066006520978, −11.41169175408440, −11.05551291742330, −10.92145866528188, −10.38328055437617, −9.514038027114732, −9.187626221616145, −8.765282584235222, −8.379648367795811, −7.782799269650101, −7.225987778990375, −6.666992690723705, −6.540722491637791, −5.729221308440553, −5.140688729050440, −4.634930692330742, −4.142731474189651, −3.795870891502588, −2.922286637358840, −2.534186664533980, −1.673350092309098, −1.322596701530001, −0.2108090053513988,
0.2108090053513988, 1.322596701530001, 1.673350092309098, 2.534186664533980, 2.922286637358840, 3.795870891502588, 4.142731474189651, 4.634930692330742, 5.140688729050440, 5.729221308440553, 6.540722491637791, 6.666992690723705, 7.225987778990375, 7.782799269650101, 8.379648367795811, 8.765282584235222, 9.187626221616145, 9.514038027114732, 10.38328055437617, 10.92145866528188, 11.05551291742330, 11.41169175408440, 12.15066006520978, 12.55193478011075, 13.04138958809108
