L(s) = 1 | − 7-s − 5·13-s + 19-s − 10·25-s − 11·31-s − 11·37-s + 13·43-s − 6·49-s − 14·61-s − 5·67-s − 17·73-s − 17·79-s + 5·91-s − 14·97-s − 26·103-s + 19·109-s − 22·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.38·13-s + 0.229·19-s − 2·25-s − 1.97·31-s − 1.80·37-s + 1.98·43-s − 6/7·49-s − 1.79·61-s − 0.610·67-s − 1.98·73-s − 1.91·79-s + 0.524·91-s − 1.42·97-s − 2.56·103-s + 1.81·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.0867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.735337316128786531026605872727, −8.671131994267711853551069541782, −7.916986045840507899519835116874, −7.47543495519043085394123869258, −7.36288626373026552746016215448, −7.18168453819984626874993426681, −6.28794468224460445429142665671, −6.23606275897716581455959955764, −5.50973610260415290305082869205, −5.42300451138715627894427055189, −4.90347528199009248676168407541, −4.25634642556941046501974163839, −3.99133648412656013929948771383, −3.52926385911289743124619338202, −2.81474833742590553500052434650, −2.62253147020274484765712305503, −1.73243019184440802996069156400, −1.54411190143584267816944787932, 0, 0,
1.54411190143584267816944787932, 1.73243019184440802996069156400, 2.62253147020274484765712305503, 2.81474833742590553500052434650, 3.52926385911289743124619338202, 3.99133648412656013929948771383, 4.25634642556941046501974163839, 4.90347528199009248676168407541, 5.42300451138715627894427055189, 5.50973610260415290305082869205, 6.23606275897716581455959955764, 6.28794468224460445429142665671, 7.18168453819984626874993426681, 7.36288626373026552746016215448, 7.47543495519043085394123869258, 7.916986045840507899519835116874, 8.671131994267711853551069541782, 8.735337316128786531026605872727