| L(s) = 1 | + 4·2-s + 4·3-s + 8·4-s + 16·6-s + 12·7-s + 4·8-s − 6·9-s − 4·11-s + 32·12-s − 8·13-s + 48·14-s − 25·16-s − 24·18-s + 48·21-s − 16·22-s + 16·24-s − 32·26-s − 40·27-s + 96·28-s + 40·29-s + 40·31-s − 56·32-s − 16·33-s − 48·36-s − 40·37-s − 32·39-s + 32·41-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4/3·3-s + 2·4-s + 8/3·6-s + 12/7·7-s + 1/2·8-s − 2/3·9-s − 0.363·11-s + 8/3·12-s − 0.615·13-s + 24/7·14-s − 1.56·16-s − 4/3·18-s + 16/7·21-s − 0.727·22-s + 2/3·24-s − 1.23·26-s − 1.48·27-s + 24/7·28-s + 1.37·29-s + 1.29·31-s − 7/4·32-s − 0.484·33-s − 4/3·36-s − 1.08·37-s − 0.820·39-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(16.45609908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.45609908\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 8 T + 8 p T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T + p^{3} T^{2} - p^{2} T^{3} - 7 T^{4} - p^{4} T^{5} + p^{7} T^{6} - p^{8} T^{7} + p^{8} T^{8} \) |
| 3 | $D_{4}$ | \( ( 1 - 2 T + p^{2} T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 744 T^{3} + 7519 T^{4} - 744 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 172 T^{3} - 2386 T^{4} + 172 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 418 T^{2} + 197763 T^{4} - 418 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 232162 T^{4} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 p T^{2} + 857778 T^{4} - 56 p^{5} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 20 T + 1532 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 39240 T^{3} + 1924322 T^{4} - 39240 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 46560 T^{3} + 2667767 T^{4} + 46560 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 57248 T^{3} + 6389378 T^{4} - 57248 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6334 T^{2} + 16708731 T^{4} - 6334 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 1736 T^{3} - 6608737 T^{4} + 1736 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 40 T + 5768 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 56 T + 1568 T^{2} + 2632 T^{3} - 12442366 T^{4} + 2632 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 148 T + 12878 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 84 T + 3528 T^{2} + 155316 T^{3} - 33332642 T^{4} + 155316 p^{2} T^{5} + 3528 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4 p T + 8 p^{2} T^{2} - 57112 p T^{3} + 322400399 T^{4} - 57112 p^{3} T^{5} + 8 p^{6} T^{6} - 4 p^{7} T^{7} + p^{8} T^{8} \) |
| 73 | $C_2^3$ | \( 1 + 18164482 T^{4} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 32 T + 11528 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 271804 T^{3} + 51880814 T^{4} - 271804 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1908200 T^{3} + 179436962 T^{4} - 1908200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 68 T + 2312 T^{2} + 801924 T^{3} - 171375106 T^{4} + 801924 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.007303158887678881878481127331, −8.006608010900120690255813664910, −7.76577134607975513720267819755, −7.56542689813259960261944558374, −7.24432699138019746793795857570, −6.73003549337497718091892551859, −6.47196969353723155583697654541, −6.37769474709651645887302985419, −6.07706939874814366964230715415, −5.80195438709580868945055962354, −5.35050802313202043170969691891, −4.98093631442900389308851800050, −4.96758523546999614660741903877, −4.92176796862328387818170088529, −4.42647944020072834815677836611, −4.12241865524905287408393833628, −3.76582587403346530847288883772, −3.37786213175208090493346441149, −3.20202560360149844982516782124, −2.75138223134574202589462586559, −2.42467087540452124327448892888, −2.39845058721585867256639806797, −1.97371474324938030590086717983, −1.16778100132164865753394403457, −0.56326310254782374094750948085,
0.56326310254782374094750948085, 1.16778100132164865753394403457, 1.97371474324938030590086717983, 2.39845058721585867256639806797, 2.42467087540452124327448892888, 2.75138223134574202589462586559, 3.20202560360149844982516782124, 3.37786213175208090493346441149, 3.76582587403346530847288883772, 4.12241865524905287408393833628, 4.42647944020072834815677836611, 4.92176796862328387818170088529, 4.96758523546999614660741903877, 4.98093631442900389308851800050, 5.35050802313202043170969691891, 5.80195438709580868945055962354, 6.07706939874814366964230715415, 6.37769474709651645887302985419, 6.47196969353723155583697654541, 6.73003549337497718091892551859, 7.24432699138019746793795857570, 7.56542689813259960261944558374, 7.76577134607975513720267819755, 8.006608010900120690255813664910, 8.007303158887678881878481127331
