| L(s) = 1 | + 8·11-s − 8·19-s + 4·29-s − 8·31-s − 4·41-s − 2·49-s + 8·59-s + 12·61-s − 32·71-s + 8·79-s + 20·89-s + 12·101-s − 28·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 1.83·19-s + 0.742·29-s − 1.43·31-s − 0.624·41-s − 2/7·49-s + 1.04·59-s + 1.53·61-s − 3.79·71-s + 0.900·79-s + 2.11·89-s + 1.19·101-s − 2.68·109-s + 2.36·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.137932632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.137932632\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.447013626811723953284927187502, −7.56671573885717328811483008118, −7.49895576293822904750990238530, −6.87450439503230850028034699206, −6.58475957569731288668142340809, −6.54399536793065646283794025886, −6.14916266953277597686497323988, −5.62385748937213260582819145840, −5.41908178405877698729983541423, −4.71261496006883292606892435988, −4.54990775960046182119127170760, −3.99308811057150348149450094140, −3.91763178484183655056617008760, −3.45766867381664674753910570465, −3.03750094885461398924599470509, −2.30023723554424436799213518326, −2.08503955715628761834504907120, −1.46189221535042755390521114735, −1.18314950788353118339021252679, −0.36608818483602426947656564574,
0.36608818483602426947656564574, 1.18314950788353118339021252679, 1.46189221535042755390521114735, 2.08503955715628761834504907120, 2.30023723554424436799213518326, 3.03750094885461398924599470509, 3.45766867381664674753910570465, 3.91763178484183655056617008760, 3.99308811057150348149450094140, 4.54990775960046182119127170760, 4.71261496006883292606892435988, 5.41908178405877698729983541423, 5.62385748937213260582819145840, 6.14916266953277597686497323988, 6.54399536793065646283794025886, 6.58475957569731288668142340809, 6.87450439503230850028034699206, 7.49895576293822904750990238530, 7.56671573885717328811483008118, 8.447013626811723953284927187502
