| L(s) = 1 | + 2·2-s + 3·4-s + 6·8-s − 9-s + 9·16-s − 2·18-s + 2·23-s + 2·29-s + 12·32-s − 3·36-s + 2·43-s + 4·46-s − 2·49-s − 2·53-s + 4·58-s + 18·64-s + 2·67-s + 2·71-s − 6·72-s − 2·79-s + 81-s + 4·86-s + 6·92-s − 4·98-s − 4·106-s − 2·109-s + 6·116-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 6·8-s − 9-s + 9·16-s − 2·18-s + 2·23-s + 2·29-s + 12·32-s − 3·36-s + 2·43-s + 4·46-s − 2·49-s − 2·53-s + 4·58-s + 18·64-s + 2·67-s + 2·71-s − 6·72-s − 2·79-s + 81-s + 4·86-s + 6·92-s − 4·98-s − 4·106-s − 2·109-s + 6·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(11.52629487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.52629487\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.34237091326633611909591400426, −6.15230391453101040127840736921, −5.89477856955535634630726249703, −5.55326166545353886670759470751, −5.25588961940768371687845900792, −5.24638965835726173611439024198, −5.04869304506502124769676075473, −5.01706190421377479453231593470, −4.57308321619919983010699535753, −4.55557331660561406484962789940, −4.44681207456185574862429486293, −4.02207271142660810895031190101, −3.83042628948605678523422784734, −3.76808270074199058938458993616, −3.26439295275036186741243136334, −3.20623443429358315264368234419, −3.12426495926518023482513496750, −2.51811657700276933781732618926, −2.45724450838127310295939510109, −2.43594612062428132372307050059, −2.15303723599166400292413236554, −1.55608067127091957716119999056, −1.31604140028686303302005269770, −1.16727694661583158975956485704, −0.985884300197015964115202877043,
0.985884300197015964115202877043, 1.16727694661583158975956485704, 1.31604140028686303302005269770, 1.55608067127091957716119999056, 2.15303723599166400292413236554, 2.43594612062428132372307050059, 2.45724450838127310295939510109, 2.51811657700276933781732618926, 3.12426495926518023482513496750, 3.20623443429358315264368234419, 3.26439295275036186741243136334, 3.76808270074199058938458993616, 3.83042628948605678523422784734, 4.02207271142660810895031190101, 4.44681207456185574862429486293, 4.55557331660561406484962789940, 4.57308321619919983010699535753, 5.01706190421377479453231593470, 5.04869304506502124769676075473, 5.24638965835726173611439024198, 5.25588961940768371687845900792, 5.55326166545353886670759470751, 5.89477856955535634630726249703, 6.15230391453101040127840736921, 6.34237091326633611909591400426
