| L(s) = 1 | + 2·2-s + 4-s − 9-s − 17-s − 2·18-s − 4·25-s − 2·32-s − 2·34-s − 36-s + 4·47-s − 49-s − 8·50-s − 2·53-s + 2·59-s − 4·64-s − 68-s + 8·83-s − 2·89-s + 8·94-s − 2·98-s − 4·100-s + 2·101-s + 2·103-s − 4·106-s + 4·118-s − 4·121-s + 127-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 9-s − 17-s − 2·18-s − 4·25-s − 2·32-s − 2·34-s − 36-s + 4·47-s − 49-s − 8·50-s − 2·53-s + 2·59-s − 4·64-s − 68-s + 8·83-s − 2·89-s + 8·94-s − 2·98-s − 4·100-s + 2·101-s + 2·103-s − 4·106-s + 4·118-s − 4·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308381307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308381307\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 47 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_1$ | \( ( 1 - T )^{8} \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−7.47261228052334532118473597486, −7.44494324473914114134352655656, −7.12765120247226943879114138473, −6.82735316295868090658360247170, −6.49927892204442259880984748444, −6.30859000117540929748295568381, −6.01147990756952397556880159451, −5.86076824090073874896331840795, −5.75878655530612182249045911703, −5.36466029016716930557473677270, −5.22943843399799874715578687605, −4.97094963084410077150673851390, −4.70331704210385831845864675399, −4.48453321000380256502910651092, −4.30614247235089430614961612031, −3.84099739825948163578429689560, −3.74545548360107902941522006729, −3.64964892789936085296986540618, −3.51475578958502495483302169726, −2.81711902218600475804408072674, −2.57705585961564947296051439386, −2.15833606164075396586359970695, −1.89740679322066617615386798472, −1.85968381314560660724426055721, −0.73193377152842458121715506240,
0.73193377152842458121715506240, 1.85968381314560660724426055721, 1.89740679322066617615386798472, 2.15833606164075396586359970695, 2.57705585961564947296051439386, 2.81711902218600475804408072674, 3.51475578958502495483302169726, 3.64964892789936085296986540618, 3.74545548360107902941522006729, 3.84099739825948163578429689560, 4.30614247235089430614961612031, 4.48453321000380256502910651092, 4.70331704210385831845864675399, 4.97094963084410077150673851390, 5.22943843399799874715578687605, 5.36466029016716930557473677270, 5.75878655530612182249045911703, 5.86076824090073874896331840795, 6.01147990756952397556880159451, 6.30859000117540929748295568381, 6.49927892204442259880984748444, 6.82735316295868090658360247170, 7.12765120247226943879114138473, 7.44494324473914114134352655656, 7.47261228052334532118473597486
