| L(s) = 1 | − 4·7-s + 2·25-s + 20·31-s − 2·49-s + 28·73-s + 20·79-s + 4·97-s − 28·103-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s − 8·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2/5·25-s + 3.59·31-s − 2/7·49-s + 3.27·73-s + 2.25·79-s + 0.406·97-s − 2.75·103-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-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 + 2·169-s + 0.0760·173-s − 0.604·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.175850264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.175850264\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−12.08278025272979029846000667134, −11.72416379557190329074753040666, −11.00723200847019826061320292563, −10.63674661622151076626270211416, −10.05913928040055511369463331455, −9.586479804041296252602956632838, −9.520030237277237139040140729588, −8.723941396503243354537087746867, −8.043749302314307753779642083378, −7.989377706501535148655064055663, −6.91634917651026700944572235287, −6.50007918929654233675800096403, −6.42453697861332577235081388122, −5.58852923218766745164308619075, −4.88202694461564893056033644935, −4.34660200005802142189931412234, −3.49605654314827524359638190567, −3.01699593577955395555041788775, −2.31131467725737623642794538331, −0.838941618298207999531843342580,
0.838941618298207999531843342580, 2.31131467725737623642794538331, 3.01699593577955395555041788775, 3.49605654314827524359638190567, 4.34660200005802142189931412234, 4.88202694461564893056033644935, 5.58852923218766745164308619075, 6.42453697861332577235081388122, 6.50007918929654233675800096403, 6.91634917651026700944572235287, 7.989377706501535148655064055663, 8.043749302314307753779642083378, 8.723941396503243354537087746867, 9.520030237277237139040140729588, 9.586479804041296252602956632838, 10.05913928040055511369463331455, 10.63674661622151076626270211416, 11.00723200847019826061320292563, 11.72416379557190329074753040666, 12.08278025272979029846000667134
