| L(s) = 1 | − 6·7-s − 3·9-s + 19-s + 3·25-s + 2·43-s + 13·49-s + 14·61-s + 18·63-s − 6·73-s + 9·81-s + 15·121-s + 127-s + 131-s − 6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 3·171-s + 173-s − 18·175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 9-s + 0.229·19-s + 3/5·25-s + 0.304·43-s + 13/7·49-s + 1.79·61-s + 2.26·63-s − 0.702·73-s + 81-s + 1.36·121-s + 0.0887·127-s + 0.0873·131-s − 0.520·133-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 − 0.229·171-s + 0.0760·173-s − 1.36·175-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 87 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.961129469380962181989895212878, −7.32305130052660723092311766345, −6.94425131620653819275823424387, −6.59760480537648271802059262796, −6.11070418133103761585213486013, −5.83615360060713533534812825033, −5.32506833441713424107768067043, −4.76444957510243367255024010339, −4.05768926443047187818056846955, −3.52186728479255860529351470583, −3.12901467819300418738126057534, −2.75614081137912399940820930647, −2.08483614218035674769636783112, −0.854302633003477795640210094843, 0,
0.854302633003477795640210094843, 2.08483614218035674769636783112, 2.75614081137912399940820930647, 3.12901467819300418738126057534, 3.52186728479255860529351470583, 4.05768926443047187818056846955, 4.76444957510243367255024010339, 5.32506833441713424107768067043, 5.83615360060713533534812825033, 6.11070418133103761585213486013, 6.59760480537648271802059262796, 6.94425131620653819275823424387, 7.32305130052660723092311766345, 7.961129469380962181989895212878
