| L(s) = 1 | + 2·4-s − 5-s + 7-s + 2·9-s + 3·16-s + 17-s − 2·19-s − 2·20-s − 23-s + 2·28-s − 35-s + 4·36-s + 43-s − 2·45-s − 47-s + 61-s + 2·63-s + 4·64-s + 2·68-s − 4·73-s − 4·76-s − 3·80-s + 3·81-s + 83-s − 85-s − 2·92-s + 2·95-s + ⋯ |
| L(s) = 1 | + 2·4-s − 5-s + 7-s + 2·9-s + 3·16-s + 17-s − 2·19-s − 2·20-s − 23-s + 2·28-s − 35-s + 4·36-s + 43-s − 2·45-s − 47-s + 61-s + 2·63-s + 4·64-s + 2·68-s − 4·73-s − 4·76-s − 3·80-s + 3·81-s + 83-s − 85-s − 2·92-s + 2·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5285401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.456008419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.456008419\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 47 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$ | \( ( 1 + T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.505558190963541515355826457240, −8.863833339189226085310249890089, −8.252595315171550841223554050854, −8.212518551529824606374717964760, −7.61747716589934629497263671869, −7.53122863053214623181693934999, −7.21082374994622585437482766891, −6.79183658768306434308469853515, −6.28305167736999195354258562142, −6.10647723588202936299945311270, −5.57146831041719926386281254435, −4.94461317535897370366222648883, −4.40215043619379944122032119069, −4.22756262192636300870587575181, −3.58884387158597050933961588854, −3.38525033073511601876356175405, −2.30407897469613411637966870891, −2.29425632982031066056024782958, −1.46625359445365050536824984124, −1.30469611812333123409243508434,
1.30469611812333123409243508434, 1.46625359445365050536824984124, 2.29425632982031066056024782958, 2.30407897469613411637966870891, 3.38525033073511601876356175405, 3.58884387158597050933961588854, 4.22756262192636300870587575181, 4.40215043619379944122032119069, 4.94461317535897370366222648883, 5.57146831041719926386281254435, 6.10647723588202936299945311270, 6.28305167736999195354258562142, 6.79183658768306434308469853515, 7.21082374994622585437482766891, 7.53122863053214623181693934999, 7.61747716589934629497263671869, 8.212518551529824606374717964760, 8.252595315171550841223554050854, 8.863833339189226085310249890089, 9.505558190963541515355826457240
