L(s) = 1 | − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯ |
L(s) = 1 | − 2·4-s − 2·5-s − 9-s + 2·13-s + 3·16-s − 2·17-s + 4·20-s + 3·25-s + 2·36-s − 2·37-s − 2·41-s + 2·45-s − 49-s − 4·52-s + 2·53-s − 4·64-s − 4·65-s + 4·68-s − 2·73-s − 6·80-s + 81-s + 4·85-s − 6·100-s + 2·113-s − 2·117-s − 121-s − 6·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07615071625\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07615071625\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−10.25696943349034234137116405243, −9.935290020569122213739987438421, −9.326942655554410349488623302761, −9.265677000862623320034465975496, −8.768357939163126364910945396664, −8.622440126044797827118706350253, −8.536369759493712591970660613850, −8.387361065421477009127377174460, −8.302147952705361882193863854075, −7.55840518286483759960699494642, −7.35518884853854989998619678137, −7.21735457249255791734349745664, −6.51577319627012882774980448639, −6.30474776983992920335379169545, −6.18483581618578829344299054578, −5.41607412047591697559067547569, −5.15770407288810986410585864163, −5.01178891049608934019955746078, −4.55972529911887285331881387887, −4.01588990366557451481704433847, −3.94693333798037946909518849036, −3.62548533295285780113509540828, −3.23664863338373525323008173673, −2.76326840611427196538532991642, −1.52908883261400833847866302720,
1.52908883261400833847866302720, 2.76326840611427196538532991642, 3.23664863338373525323008173673, 3.62548533295285780113509540828, 3.94693333798037946909518849036, 4.01588990366557451481704433847, 4.55972529911887285331881387887, 5.01178891049608934019955746078, 5.15770407288810986410585864163, 5.41607412047591697559067547569, 6.18483581618578829344299054578, 6.30474776983992920335379169545, 6.51577319627012882774980448639, 7.21735457249255791734349745664, 7.35518884853854989998619678137, 7.55840518286483759960699494642, 8.302147952705361882193863854075, 8.387361065421477009127377174460, 8.536369759493712591970660613850, 8.622440126044797827118706350253, 8.768357939163126364910945396664, 9.265677000862623320034465975496, 9.326942655554410349488623302761, 9.935290020569122213739987438421, 10.25696943349034234137116405243