L(s) = 1 | + 4·3-s + 6·9-s − 4·27-s − 12·41-s + 20·43-s + 14·49-s + 28·67-s − 37·81-s + 36·83-s − 36·89-s + 12·107-s − 14·121-s − 48·123-s + 127-s + 80·129-s + 131-s + 137-s + 139-s + 56·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·9-s − 0.769·27-s − 1.87·41-s + 3.04·43-s + 2·49-s + 3.42·67-s − 4.11·81-s + 3.95·83-s − 3.81·89-s + 1.16·107-s − 1.27·121-s − 4.32·123-s + 0.0887·127-s + 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.61·147-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 + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.075994773\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.075994773\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.534829036418252700257629357443, −9.043286683129683219459596029754, −8.730054488693157569360021493151, −8.591579832571673505528003084598, −8.019838620753455863560859146050, −7.82336287633584810161167813566, −7.38715341089691132133790223606, −7.02556548992560164162144608904, −6.49255958603362394750607504820, −5.96858853173716671788493034377, −5.41852473123892389915316951896, −5.14925602938534690765238908263, −4.29569294913590630432367766231, −3.89753377453048086960262528029, −3.64768968698289207823728794986, −3.08735359786260095590437152702, −2.43016051313926583327314117215, −2.42481699904604462119437761230, −1.71493873068809262695085591796, −0.74244896642002666158509946344,
0.74244896642002666158509946344, 1.71493873068809262695085591796, 2.42481699904604462119437761230, 2.43016051313926583327314117215, 3.08735359786260095590437152702, 3.64768968698289207823728794986, 3.89753377453048086960262528029, 4.29569294913590630432367766231, 5.14925602938534690765238908263, 5.41852473123892389915316951896, 5.96858853173716671788493034377, 6.49255958603362394750607504820, 7.02556548992560164162144608904, 7.38715341089691132133790223606, 7.82336287633584810161167813566, 8.019838620753455863560859146050, 8.591579832571673505528003084598, 8.730054488693157569360021493151, 9.043286683129683219459596029754, 9.534829036418252700257629357443