| L(s) = 1 | + 2·9-s − 4·17-s − 6·25-s + 16·31-s − 20·41-s − 16·47-s + 16·71-s + 12·73-s + 16·79-s − 5·81-s − 20·89-s − 4·97-s + 32·103-s + 4·113-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 0.970·17-s − 6/5·25-s + 2.87·31-s − 3.12·41-s − 2.33·47-s + 1.89·71-s + 1.40·73-s + 1.80·79-s − 5/9·81-s − 2.11·89-s − 0.406·97-s + 3.15·103-s + 0.376·113-s + 1.63·121-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 − 0.646·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.869415734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869415734\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−9.337740345186374845700511101938, −8.625126234669142390965667685909, −8.192702098117068456068239506218, −7.86059166450488347634678335551, −7.37465255585159860174696195791, −6.86135303238179194493046585279, −6.59266702105441550867394730044, −6.39972437515509583460445999502, −6.00231656089656798838812285196, −5.21003215360428475014748173103, −5.09887299327169344857220318140, −4.46388080741141509418774897393, −4.45767005383746075494024577986, −3.62826074722234604933079682280, −3.40653435768586061531216852975, −2.88342213822445727805302303955, −2.08519459010180559264408081605, −1.96613803356982411648777077716, −1.21767964594959544451573829636, −0.44147507189050015585369494216,
0.44147507189050015585369494216, 1.21767964594959544451573829636, 1.96613803356982411648777077716, 2.08519459010180559264408081605, 2.88342213822445727805302303955, 3.40653435768586061531216852975, 3.62826074722234604933079682280, 4.45767005383746075494024577986, 4.46388080741141509418774897393, 5.09887299327169344857220318140, 5.21003215360428475014748173103, 6.00231656089656798838812285196, 6.39972437515509583460445999502, 6.59266702105441550867394730044, 6.86135303238179194493046585279, 7.37465255585159860174696195791, 7.86059166450488347634678335551, 8.192702098117068456068239506218, 8.625126234669142390965667685909, 9.337740345186374845700511101938
