| L(s) = 1 | − 4·3-s − 8·7-s + 8·9-s + 32·21-s − 12·27-s − 32·31-s − 32·37-s − 8·43-s + 32·49-s − 24·61-s − 64·63-s − 24·67-s − 32·73-s + 23·81-s + 128·93-s + 32·97-s − 24·103-s + 128·111-s − 20·121-s + 127-s + 32·129-s + 131-s + 137-s + 139-s − 128·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.02·7-s + 8/3·9-s + 6.98·21-s − 2.30·27-s − 5.74·31-s − 5.26·37-s − 1.21·43-s + 32/7·49-s − 3.07·61-s − 8.06·63-s − 2.93·67-s − 3.74·73-s + 23/9·81-s + 13.2·93-s + 3.24·97-s − 2.36·103-s + 12.1·111-s − 1.81·121-s + 0.0887·127-s + 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 958 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 3682 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 3854 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 12878 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.33141066806559101920320673994, −7.08865658340810206136314088578, −6.90312460301925017665630973448, −6.64770209205872827267057240311, −6.59456570505574244109479947746, −6.14338251429471009493355995330, −5.98950487083743241321939682913, −5.95040283240010447057077064634, −5.79472229497362126952728517986, −5.47889022127029328100287014697, −5.16720725357775466167810705883, −5.10517222830260620554518905651, −4.95349925205421325153156435037, −4.60051738945155253587241994299, −4.08462660640692026455977735693, −3.91058196048793774274400447395, −3.59892778277536946967825673883, −3.54269177073260169902359654027, −3.32330537809552337108992570988, −2.95623639703052422194080173491, −2.88691164004279061749837348432, −2.08122662945271656368633171479, −1.83313232375084836687615780898, −1.55996913568475495543508443069, −1.33915074520200397122627638469, 0, 0, 0, 0,
1.33915074520200397122627638469, 1.55996913568475495543508443069, 1.83313232375084836687615780898, 2.08122662945271656368633171479, 2.88691164004279061749837348432, 2.95623639703052422194080173491, 3.32330537809552337108992570988, 3.54269177073260169902359654027, 3.59892778277536946967825673883, 3.91058196048793774274400447395, 4.08462660640692026455977735693, 4.60051738945155253587241994299, 4.95349925205421325153156435037, 5.10517222830260620554518905651, 5.16720725357775466167810705883, 5.47889022127029328100287014697, 5.79472229497362126952728517986, 5.95040283240010447057077064634, 5.98950487083743241321939682913, 6.14338251429471009493355995330, 6.59456570505574244109479947746, 6.64770209205872827267057240311, 6.90312460301925017665630973448, 7.08865658340810206136314088578, 7.33141066806559101920320673994
