| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 8·21-s + 4·29-s + 4·41-s − 4·43-s + 2·47-s + 6·49-s − 2·61-s + 4·63-s − 2·67-s − 8·87-s + 4·101-s + 4·103-s + 2·107-s + 2·109-s + 2·121-s − 8·123-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 4·141-s − 12·147-s + 149-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4·7-s + 9-s − 8·21-s + 4·29-s + 4·41-s − 4·43-s + 2·47-s + 6·49-s − 2·61-s + 4·63-s − 2·67-s − 8·87-s + 4·101-s + 4·103-s + 2·107-s + 2·109-s + 2·121-s − 8·123-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 4·141-s − 12·147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.734205863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734205863\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 47 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 53 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.12559588596155914759705362497, −5.90806520187116067001145941268, −5.70196599494598929383165716967, −5.62084944833434498516951877002, −5.47923043703536918048730434549, −4.96504976219949857300362653745, −4.81249691252896212866574397069, −4.78882258062993015213640737377, −4.70069323542054278298463675761, −4.54882166413855651252357203775, −4.54452971770920375037687192289, −4.26551612399996911637692952806, −3.84446131860016018738678270684, −3.48592615637549310945588258206, −3.20684500268021427538012663884, −3.05996549815422088108326301628, −2.95431098526613991762991281851, −2.28672155570065675977511894169, −2.26584819545634445180300634459, −1.98478286030508818142116350718, −1.77439454314597711757064741245, −1.51214057368935250304377143759, −1.07904556784875570062588343217, −0.909903505721793999753753277241, −0.68376958510417186764408494987,
0.68376958510417186764408494987, 0.909903505721793999753753277241, 1.07904556784875570062588343217, 1.51214057368935250304377143759, 1.77439454314597711757064741245, 1.98478286030508818142116350718, 2.26584819545634445180300634459, 2.28672155570065675977511894169, 2.95431098526613991762991281851, 3.05996549815422088108326301628, 3.20684500268021427538012663884, 3.48592615637549310945588258206, 3.84446131860016018738678270684, 4.26551612399996911637692952806, 4.54452971770920375037687192289, 4.54882166413855651252357203775, 4.70069323542054278298463675761, 4.78882258062993015213640737377, 4.81249691252896212866574397069, 4.96504976219949857300362653745, 5.47923043703536918048730434549, 5.62084944833434498516951877002, 5.70196599494598929383165716967, 5.90806520187116067001145941268, 6.12559588596155914759705362497
