| L(s) = 1 | + 3·5-s − 7-s + 7·13-s − 3·17-s + 2·19-s − 4·23-s + 4·25-s − 7·29-s + 10·31-s − 3·35-s + 37-s + 5·41-s − 6·43-s − 6·47-s + 49-s + 5·53-s − 6·59-s + 10·61-s + 21·65-s + 8·67-s − 10·71-s + 10·73-s + 2·79-s − 16·83-s − 9·85-s − 3·89-s − 7·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 1.94·13-s − 0.727·17-s + 0.458·19-s − 0.834·23-s + 4/5·25-s − 1.29·29-s + 1.79·31-s − 0.507·35-s + 0.164·37-s + 0.780·41-s − 0.914·43-s − 0.875·47-s + 1/7·49-s + 0.686·53-s − 0.781·59-s + 1.28·61-s + 2.60·65-s + 0.977·67-s − 1.18·71-s + 1.17·73-s + 0.225·79-s − 1.75·83-s − 0.976·85-s − 0.317·89-s − 0.733·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.606324066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606324066\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
| 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
−13.56792814201412, −13.09547073683620, −12.91753329288394, −12.08866627625526, −11.48337958798043, −11.18591926416123, −10.55778420050972, −10.11556652202449, −9.601989808663826, −9.318833660224684, −8.586886071062793, −8.330258641122646, −7.684753494380421, −6.813183585674746, −6.474625443536137, −6.088049191116146, −5.575498200598281, −5.156496577513660, −4.167122673939707, −3.927724530799940, −3.084955090847805, −2.580674719475272, −1.801048457598328, −1.405918976592454, −0.5766720674414525,
0.5766720674414525, 1.405918976592454, 1.801048457598328, 2.580674719475272, 3.084955090847805, 3.927724530799940, 4.167122673939707, 5.156496577513660, 5.575498200598281, 6.088049191116146, 6.474625443536137, 6.813183585674746, 7.684753494380421, 8.330258641122646, 8.586886071062793, 9.318833660224684, 9.601989808663826, 10.11556652202449, 10.55778420050972, 11.18591926416123, 11.48337958798043, 12.08866627625526, 12.91753329288394, 13.09547073683620, 13.56792814201412
