| L(s) = 1 | − 5-s + 6·9-s − 6·11-s − 2·19-s + 5·25-s − 2·29-s + 2·31-s + 30·41-s − 6·45-s + 6·55-s + 2·59-s + 24·61-s + 12·71-s − 14·79-s + 9·81-s − 28·89-s + 2·95-s − 36·99-s + 24·101-s + 16·109-s + 7·121-s − 14·125-s + 127-s + 131-s + 137-s + 139-s + 2·145-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 2·9-s − 1.80·11-s − 0.458·19-s + 25-s − 0.371·29-s + 0.359·31-s + 4.68·41-s − 0.894·45-s + 0.809·55-s + 0.260·59-s + 3.07·61-s + 1.42·71-s − 1.57·79-s + 81-s − 2.96·89-s + 0.205·95-s − 3.61·99-s + 2.38·101-s + 1.53·109-s + 7/11·121-s − 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.166·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.716221407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.716221407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 956 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 17 T^{2} + 1656 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 9032 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 9396 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - T + 104 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 12 T + 101 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 62 T^{2} + 1731 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 25308 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7 T + 156 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.25394571343315023491584726957, −7.12266627363024209208745554155, −6.65846745887568499781753674940, −6.58796711506275880962324399771, −6.23353756148679992909265695095, −6.21704850716647199715687633832, −5.64157732044416580378530754101, −5.43388114629236229923074894758, −5.34291805562788490287603599609, −5.31589916978674769057325598316, −4.61451230414475539292835242962, −4.56363595166262547617027223139, −4.41499023462223493689252778271, −4.03944148425413741050640058134, −3.96066028588152362994559450636, −3.81604207150010575292110028939, −3.09914533081311057246613997630, −2.91229851661542080492586322657, −2.83701511865367613755596623688, −2.39015963228159148563648643042, −1.97147768849012565373514973959, −1.94149152314632860883411906680, −1.25118665028775341746429106589, −0.70309097798995950529974344548, −0.67009093197832453107159192149,
0.67009093197832453107159192149, 0.70309097798995950529974344548, 1.25118665028775341746429106589, 1.94149152314632860883411906680, 1.97147768849012565373514973959, 2.39015963228159148563648643042, 2.83701511865367613755596623688, 2.91229851661542080492586322657, 3.09914533081311057246613997630, 3.81604207150010575292110028939, 3.96066028588152362994559450636, 4.03944148425413741050640058134, 4.41499023462223493689252778271, 4.56363595166262547617027223139, 4.61451230414475539292835242962, 5.31589916978674769057325598316, 5.34291805562788490287603599609, 5.43388114629236229923074894758, 5.64157732044416580378530754101, 6.21704850716647199715687633832, 6.23353756148679992909265695095, 6.58796711506275880962324399771, 6.65846745887568499781753674940, 7.12266627363024209208745554155, 7.25394571343315023491584726957
