| L(s) = 1 | + 2·9-s − 8·19-s − 12·29-s + 8·31-s + 12·41-s + 10·49-s + 24·59-s + 4·61-s + 24·71-s + 16·79-s − 5·81-s + 12·89-s + 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 16·171-s + 173-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 1.83·19-s − 2.22·29-s + 1.43·31-s + 1.87·41-s + 10/7·49-s + 3.12·59-s + 0.512·61-s + 2.84·71-s + 1.80·79-s − 5/9·81-s + 1.27·89-s + 1.19·101-s − 0.383·109-s − 2·121-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 + 1.69·169-s − 1.22·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.595418988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595418988\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−11.32734545736365554889082141083, −11.11145598250743589463813941637, −10.59635395971337236071032249350, −10.11479236960394371978330958967, −9.799071690705238471210680610521, −9.109737120649100099728257769833, −8.909843331188800465642844077057, −8.214104276581169430209889968862, −7.82734651441665781875905466194, −7.34582026778733716604996637182, −6.67727215748026356498706456766, −6.45698248089948531844922588478, −5.72290744922611099450700787730, −5.26127125975652623331887073362, −4.55694476169128296686606688384, −3.87648931552195386388538551592, −3.77527774980063094396502606883, −2.37040898097720189295123822923, −2.20514001477767297587504975781, −0.857840239457185337066244292070,
0.857840239457185337066244292070, 2.20514001477767297587504975781, 2.37040898097720189295123822923, 3.77527774980063094396502606883, 3.87648931552195386388538551592, 4.55694476169128296686606688384, 5.26127125975652623331887073362, 5.72290744922611099450700787730, 6.45698248089948531844922588478, 6.67727215748026356498706456766, 7.34582026778733716604996637182, 7.82734651441665781875905466194, 8.214104276581169430209889968862, 8.909843331188800465642844077057, 9.109737120649100099728257769833, 9.799071690705238471210680610521, 10.11479236960394371978330958967, 10.59635395971337236071032249350, 11.11145598250743589463813941637, 11.32734545736365554889082141083
