| L(s) = 1 | + 5-s − 6·11-s − 4·13-s − 3·17-s − 2·19-s + 2·23-s − 4·25-s + 6·29-s − 4·31-s + 5·37-s − 5·41-s + 9·43-s + 5·47-s − 6·53-s − 6·55-s − 7·59-s − 14·61-s − 4·65-s + 12·67-s − 8·71-s + 10·73-s + 5·79-s + 11·83-s − 3·85-s − 6·89-s − 2·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.80·11-s − 1.10·13-s − 0.727·17-s − 0.458·19-s + 0.417·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s + 0.821·37-s − 0.780·41-s + 1.37·43-s + 0.729·47-s − 0.824·53-s − 0.809·55-s − 0.911·59-s − 1.79·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.562·79-s + 1.20·83-s − 0.325·85-s − 0.635·89-s − 0.205·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177454218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177454218\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−16.67137181824853, −15.82446787690564, −15.44806693395804, −15.00802487440107, −14.10329803511553, −13.76497716208068, −13.03618168130020, −12.63238723539260, −12.09620823651330, −11.14527214481445, −10.70364699940183, −10.17880682638238, −9.528645756530929, −8.982575567795437, −8.067368276116630, −7.699654108998183, −6.994977156777250, −6.192600557279008, −5.555146740720032, −4.860281580006240, −4.392419051634565, −3.173191266011753, −2.489208068701377, −1.964593296859792, −0.4740815076558818,
0.4740815076558818, 1.964593296859792, 2.489208068701377, 3.173191266011753, 4.392419051634565, 4.860281580006240, 5.555146740720032, 6.192600557279008, 6.994977156777250, 7.699654108998183, 8.067368276116630, 8.982575567795437, 9.528645756530929, 10.17880682638238, 10.70364699940183, 11.14527214481445, 12.09620823651330, 12.63238723539260, 13.03618168130020, 13.76497716208068, 14.10329803511553, 15.00802487440107, 15.44806693395804, 15.82446787690564, 16.67137181824853
