| L(s) = 1 | + 4·11-s + 12·23-s − 2·25-s + 8·29-s − 4·37-s + 8·43-s − 24·53-s + 24·67-s − 12·71-s − 16·79-s + 36·107-s + 28·109-s + 16·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 2.50·23-s − 2/5·25-s + 1.48·29-s − 0.657·37-s + 1.21·43-s − 3.29·53-s + 2.93·67-s − 1.42·71-s − 1.80·79-s + 3.48·107-s + 2.68·109-s + 1.50·113-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 + 6/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.257158912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.257158912\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−8.134013491764567981270967444633, −7.73444348375358982236853316986, −7.22950915105850518195531771543, −7.21813511860313261145244652851, −6.59554341833355144531050958602, −6.48018171309802687877965183634, −6.14411778275009514308821272797, −5.65472479360835775346575654376, −5.28484912889112647816858318845, −4.80754755102353602349826089043, −4.54711827913441491712585287685, −4.34689715481470739738476293027, −3.62131189704827098982604408522, −3.38364402230476885099865176310, −2.95197631101232213349454454254, −2.67938830523706588195711504339, −1.81984183259206743301132092386, −1.66687144577774356093778155691, −0.887945462437249440146770848193, −0.64330298196379591978131894412,
0.64330298196379591978131894412, 0.887945462437249440146770848193, 1.66687144577774356093778155691, 1.81984183259206743301132092386, 2.67938830523706588195711504339, 2.95197631101232213349454454254, 3.38364402230476885099865176310, 3.62131189704827098982604408522, 4.34689715481470739738476293027, 4.54711827913441491712585287685, 4.80754755102353602349826089043, 5.28484912889112647816858318845, 5.65472479360835775346575654376, 6.14411778275009514308821272797, 6.48018171309802687877965183634, 6.59554341833355144531050958602, 7.21813511860313261145244652851, 7.22950915105850518195531771543, 7.73444348375358982236853316986, 8.134013491764567981270967444633
