| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s + 9-s − 4·11-s − 12-s + 2·14-s + 17-s − 18-s − 4·19-s + 2·21-s + 4·22-s + 4·25-s − 2·28-s − 31-s + 32-s + 4·33-s − 34-s + 36-s + 4·38-s + 2·41-s − 2·42-s + 43-s − 4·44-s − 2·47-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s + 9-s − 4·11-s − 12-s + 2·14-s + 17-s − 18-s − 4·19-s + 2·21-s + 4·22-s + 4·25-s − 2·28-s − 31-s + 32-s + 4·33-s − 34-s + 36-s + 4·38-s + 2·41-s − 2·42-s + 43-s − 4·44-s − 2·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{4} \cdot 53^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{4} \cdot 53^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1788827874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1788827874\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| 53 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 47 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−7.36721352837928247452789200454, −7.25526983021641701715528424723, −6.60543377839777227043721311575, −6.58846416917166390814189125386, −6.47556091261835828505014061708, −6.45183198892482634946613368328, −6.39839604191415366308716762179, −5.71455058890399081663636761234, −5.66069396734535807580656999984, −5.10552245617588627620445850720, −5.08618126611087567532827400819, −5.01642236453366142014975893566, −4.77110176971373811560649476931, −4.38671743582863626729462920511, −4.19886171398229607700753855996, −3.68100823279063905551200740435, −3.32739214231538108977160603918, −3.17780648428097995183042639548, −2.95417633293786633223732077544, −2.37335633005968259184768689196, −2.34706136924791700353305827055, −2.34157223313369291689987242050, −1.64113521057914703055272674228, −0.946708770102420652876092431076, −0.50405406403653390671022060118,
0.50405406403653390671022060118, 0.946708770102420652876092431076, 1.64113521057914703055272674228, 2.34157223313369291689987242050, 2.34706136924791700353305827055, 2.37335633005968259184768689196, 2.95417633293786633223732077544, 3.17780648428097995183042639548, 3.32739214231538108977160603918, 3.68100823279063905551200740435, 4.19886171398229607700753855996, 4.38671743582863626729462920511, 4.77110176971373811560649476931, 5.01642236453366142014975893566, 5.08618126611087567532827400819, 5.10552245617588627620445850720, 5.66069396734535807580656999984, 5.71455058890399081663636761234, 6.39839604191415366308716762179, 6.45183198892482634946613368328, 6.47556091261835828505014061708, 6.58846416917166390814189125386, 6.60543377839777227043721311575, 7.25526983021641701715528424723, 7.36721352837928247452789200454
