| L(s) = 1 | − 12·5-s − 3·9-s − 12·13-s − 12·17-s + 74·25-s − 24·29-s − 6·41-s + 36·45-s + 14·49-s − 24·61-s + 144·65-s − 28·73-s + 144·85-s − 24·89-s − 2·97-s − 12·101-s − 12·113-s + 36·117-s − 13·121-s − 312·125-s + 127-s + 131-s + 137-s + 139-s + 288·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 5.36·5-s − 9-s − 3.32·13-s − 2.91·17-s + 74/5·25-s − 4.45·29-s − 0.937·41-s + 5.36·45-s + 2·49-s − 3.07·61-s + 17.8·65-s − 3.27·73-s + 15.6·85-s − 2.54·89-s − 0.203·97-s − 1.19·101-s − 1.12·113-s + 3.32·117-s − 1.18·121-s − 27.9·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 23.9·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 13 T^{2} + 48 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 34 T^{2} + 627 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 14 T^{2} - 765 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} ) \) |
| 47 | $C_2^3$ | \( 1 - 82 T^{2} + 4515 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \) |
| 83 | $C_2^3$ | \( 1 + 22 T^{2} - 6405 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.124591698039661859600404739515, −7.918552026085536196855736933449, −7.63582876511961340394320719664, −7.49256241709145088993562027461, −7.42957926624510264170362860462, −7.06630781114613986281135786900, −7.04756752554149345653807134244, −6.88494814902222667198903528682, −6.63033166851979359786595714146, −5.84623838798681336516182694221, −5.81632286740392282582301746325, −5.46361424143523309528158235011, −5.15063626370021782625474960914, −4.73708184299782208476637005324, −4.51619370223892243623190843031, −4.41295466747177788401540902775, −4.23637214690129014157014370166, −4.04108561931822478958779496458, −3.70166490465282629586641782423, −3.33181578969645458223843958720, −3.28671963971660900691154299307, −2.69919839733461288637486506592, −2.52334053218363177916110751277, −2.19416441843470999578051488556, −1.48968457857062842048663951013, 0, 0, 0, 0,
1.48968457857062842048663951013, 2.19416441843470999578051488556, 2.52334053218363177916110751277, 2.69919839733461288637486506592, 3.28671963971660900691154299307, 3.33181578969645458223843958720, 3.70166490465282629586641782423, 4.04108561931822478958779496458, 4.23637214690129014157014370166, 4.41295466747177788401540902775, 4.51619370223892243623190843031, 4.73708184299782208476637005324, 5.15063626370021782625474960914, 5.46361424143523309528158235011, 5.81632286740392282582301746325, 5.84623838798681336516182694221, 6.63033166851979359786595714146, 6.88494814902222667198903528682, 7.04756752554149345653807134244, 7.06630781114613986281135786900, 7.42957926624510264170362860462, 7.49256241709145088993562027461, 7.63582876511961340394320719664, 7.918552026085536196855736933449, 8.124591698039661859600404739515
