| L(s) = 1 | + 2·4-s − 3·5-s − 7-s − 6·11-s + 4·13-s + 6·17-s + 4·19-s − 6·20-s + 6·23-s + 5·25-s − 2·28-s + 6·29-s + 4·31-s + 3·35-s − 14·37-s + 3·41-s + 43-s − 12·44-s − 9·47-s + 8·52-s − 12·53-s + 18·55-s − 9·59-s + 10·61-s − 8·64-s − 12·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 4-s − 1.34·5-s − 0.377·7-s − 1.80·11-s + 1.10·13-s + 1.45·17-s + 0.917·19-s − 1.34·20-s + 1.25·23-s + 25-s − 0.377·28-s + 1.11·29-s + 0.718·31-s + 0.507·35-s − 2.30·37-s + 0.468·41-s + 0.152·43-s − 1.80·44-s − 1.31·47-s + 1.10·52-s − 1.64·53-s + 2.42·55-s − 1.17·59-s + 1.28·61-s − 64-s − 1.48·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.550881719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550881719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23721447998191027894763036676, −10.66313753477651668000899172835, −10.14787290235565816723462706864, −9.911675698883979559593010262779, −9.099273988875943036084657873562, −8.662843549428251733275191575752, −8.133035033046852011972786534416, −7.76395549332200421498088284238, −7.58259929619186551429954865717, −6.89628785046799844818142822052, −6.61044088810845882575608749549, −6.02882517948897863053223916508, −5.19177877508944566772636979455, −5.14594832631372341846753760086, −4.36345797875637168504452204130, −3.40270274904023974509476983168, −3.20346976534584682402833236493, −2.85040550248968902707374092413, −1.74278280528179097782973885708, −0.72277702127574534155772472532,
0.72277702127574534155772472532, 1.74278280528179097782973885708, 2.85040550248968902707374092413, 3.20346976534584682402833236493, 3.40270274904023974509476983168, 4.36345797875637168504452204130, 5.14594832631372341846753760086, 5.19177877508944566772636979455, 6.02882517948897863053223916508, 6.61044088810845882575608749549, 6.89628785046799844818142822052, 7.58259929619186551429954865717, 7.76395549332200421498088284238, 8.133035033046852011972786534416, 8.662843549428251733275191575752, 9.099273988875943036084657873562, 9.911675698883979559593010262779, 10.14787290235565816723462706864, 10.66313753477651668000899172835, 11.23721447998191027894763036676
