| L(s) = 1 | + 4-s + 2·5-s + 8·13-s + 2·20-s + 2·25-s − 4·29-s − 2·37-s + 4·41-s + 8·52-s − 2·61-s − 64-s + 16·65-s + 2·73-s + 81-s + 4·89-s − 4·97-s + 2·100-s − 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s − 2·148-s + ⋯ |
| L(s) = 1 | + 4-s + 2·5-s + 8·13-s + 2·20-s + 2·25-s − 4·29-s − 2·37-s + 4·41-s + 8·52-s − 2·61-s − 64-s + 16·65-s + 2·73-s + 81-s + 4·89-s − 4·97-s + 2·100-s − 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s − 8·145-s − 2·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(6.482892037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.482892037\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$ | \( ( 1 - T )^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.29297238121707521700936330877, −6.23298662726150483540466471606, −5.86357781332091742982085489698, −5.65296988350534869298472054807, −5.58710324861576035781463989279, −5.57187666913145199100515301077, −5.46136599186621778450574971078, −5.11301011010912190998462233852, −4.64947582485251837650653996045, −4.23651535896121596928196851308, −3.99545074357470701238648966509, −3.94570576014562021581925710889, −3.90134140255946095315776947519, −3.56252641311759928764043108461, −3.52462170581088793703702568455, −3.09752956401576953025995233556, −3.09168935815191782732982708544, −2.49724869189337234910806714464, −2.25003092462770182534077476561, −2.23577455025725957721803482243, −1.57587011085967825806985705991, −1.53683453102130554480168920674, −1.44995098844442317643185474940, −1.11758881126011125926257269677, −1.11293147012693427219868294174,
1.11293147012693427219868294174, 1.11758881126011125926257269677, 1.44995098844442317643185474940, 1.53683453102130554480168920674, 1.57587011085967825806985705991, 2.23577455025725957721803482243, 2.25003092462770182534077476561, 2.49724869189337234910806714464, 3.09168935815191782732982708544, 3.09752956401576953025995233556, 3.52462170581088793703702568455, 3.56252641311759928764043108461, 3.90134140255946095315776947519, 3.94570576014562021581925710889, 3.99545074357470701238648966509, 4.23651535896121596928196851308, 4.64947582485251837650653996045, 5.11301011010912190998462233852, 5.46136599186621778450574971078, 5.57187666913145199100515301077, 5.58710324861576035781463989279, 5.65296988350534869298472054807, 5.86357781332091742982085489698, 6.23298662726150483540466471606, 6.29297238121707521700936330877
