| L(s) = 1 | + 2·3-s − 3·4-s − 4·7-s + 9-s − 6·12-s + 2·13-s + 5·16-s + 12·19-s − 8·21-s − 9·25-s − 4·27-s + 12·28-s − 4·31-s − 3·36-s − 6·37-s + 4·39-s + 10·48-s − 2·49-s − 6·52-s + 24·57-s + 12·61-s − 4·63-s − 3·64-s + 4·67-s − 4·73-s − 18·75-s − 36·76-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 3/2·4-s − 1.51·7-s + 1/3·9-s − 1.73·12-s + 0.554·13-s + 5/4·16-s + 2.75·19-s − 1.74·21-s − 9/5·25-s − 0.769·27-s + 2.26·28-s − 0.718·31-s − 1/2·36-s − 0.986·37-s + 0.640·39-s + 1.44·48-s − 2/7·49-s − 0.832·52-s + 3.17·57-s + 1.53·61-s − 0.503·63-s − 3/8·64-s + 0.488·67-s − 0.468·73-s − 2.07·75-s − 4.12·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.182457967238804819066801847611, −8.731578058982900025567463262875, −8.356936232923193217562617522792, −7.69035551562908784013462192082, −7.43742239849917419501134588307, −6.72221402891887535126565651102, −6.02373904066748853888838701833, −5.38415572352897256696127903868, −5.21092257719357715807199072115, −3.96049656444482134108637352404, −3.84177902362047353880486406829, −3.26020657517957887232980598752, −2.75274084463573300677827364197, −1.45371564777531463821755560879, 0,
1.45371564777531463821755560879, 2.75274084463573300677827364197, 3.26020657517957887232980598752, 3.84177902362047353880486406829, 3.96049656444482134108637352404, 5.21092257719357715807199072115, 5.38415572352897256696127903868, 6.02373904066748853888838701833, 6.72221402891887535126565651102, 7.43742239849917419501134588307, 7.69035551562908784013462192082, 8.356936232923193217562617522792, 8.731578058982900025567463262875, 9.182457967238804819066801847611
