| L(s) = 1 | + 648·17-s + 2.37e3·25-s − 7.56e3·41-s − 2.68e3·49-s + 1.10e4·73-s − 9.72e3·89-s + 2.98e4·97-s + 2.80e3·113-s − 3.53e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.09e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.24·17-s + 3.79·25-s − 4.49·41-s − 1.11·49-s + 2.06·73-s − 1.22·89-s + 3.16·97-s + 0.219·113-s − 2.41·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.732·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + 2.57e−5·197-s + 2.52e−5·199-s + 2.24e−5·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.679045326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679045326\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 1186 T^{2} + p^{8} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 1342 T^{2} + p^{8} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 146 p^{2} T^{2} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 10466 T^{2} + p^{8} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 162 T + p^{4} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 66242 T^{2} + p^{8} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 62018 T^{2} + p^{8} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 285854 T^{2} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1453826 T^{2} + p^{8} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 1462178 T^{2} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 1890 T + p^{4} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 1630462 T^{2} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 7768706 T^{2} + p^{8} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 11876386 T^{2} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 19112066 T^{2} + p^{8} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 22046306 T^{2} + p^{8} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 37495106 T^{2} + p^{8} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 9393982 T^{2} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2750 T + p^{4} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 13977986 T^{2} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 7728578 T^{2} + p^{8} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2430 T + p^{4} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7454 T + p^{4} T^{2} )^{4} \) |
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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
−6.43014329700367344424045704164, −6.39674219921815140614559443301, −5.97273105661987094224447220645, −5.62840809001639329001852823898, −5.41024345491123009443467511594, −5.32871899809088431215329265114, −5.19224436626559963113623584177, −4.82098754982617094350144880767, −4.66741959712950346353203472100, −4.45249123042718133300085656995, −4.29382576535012247882016409465, −3.64177314971665110393449879724, −3.45717730873494792467102620800, −3.36485861475174179310827934315, −3.19783236011804794835051004916, −2.99524845775836148475480706011, −2.76590957787220486317759309251, −2.10134331698853130704699772140, −2.06104512877774804684490008831, −1.78213705990231868549985650750, −1.25858276863190781376410059655, −1.13533348100826526127121957511, −0.899375656574012555344506807654, −0.61278813832227631229248027420, −0.12113231724545790396343909530,
0.12113231724545790396343909530, 0.61278813832227631229248027420, 0.899375656574012555344506807654, 1.13533348100826526127121957511, 1.25858276863190781376410059655, 1.78213705990231868549985650750, 2.06104512877774804684490008831, 2.10134331698853130704699772140, 2.76590957787220486317759309251, 2.99524845775836148475480706011, 3.19783236011804794835051004916, 3.36485861475174179310827934315, 3.45717730873494792467102620800, 3.64177314971665110393449879724, 4.29382576535012247882016409465, 4.45249123042718133300085656995, 4.66741959712950346353203472100, 4.82098754982617094350144880767, 5.19224436626559963113623584177, 5.32871899809088431215329265114, 5.41024345491123009443467511594, 5.62840809001639329001852823898, 5.97273105661987094224447220645, 6.39674219921815140614559443301, 6.43014329700367344424045704164
