| L(s) = 1 | − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s − 4·16-s + 20·18-s + 16·20-s − 2·23-s − 7·25-s + 14·31-s + 32·32-s − 40·36-s + 6·37-s − 16·40-s − 8·41-s − 12·43-s − 10·45-s + 8·46-s − 10·49-s + 28·50-s + 10·59-s + 24·61-s − 56·62-s − 64·64-s + 40·72-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s − 16-s + 4.71·18-s + 3.57·20-s − 0.417·23-s − 7/5·25-s + 2.51·31-s + 5.65·32-s − 6.66·36-s + 0.986·37-s − 2.52·40-s − 1.24·41-s − 1.82·43-s − 1.49·45-s + 1.17·46-s − 1.42·49-s + 3.95·50-s + 1.30·59-s + 3.07·61-s − 7.11·62-s − 8·64-s + 4.71·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203401 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.603539619290756001226038948684, −8.533161498536631865986006945074, −8.168656713206124872277685368931, −7.896723878259049090482553992495, −7.12440344899974255905087582496, −6.49786531494913645533761715591, −6.36261389471308870138602900888, −5.55848905589213998501720184887, −5.03806026965038014721162628365, −4.25784277552826557251982155070, −3.23825086755486375020827984391, −2.34752016888707598740907779118, −2.04157827948359597493893608081, −1.02574592709347212312642397395, 0,
1.02574592709347212312642397395, 2.04157827948359597493893608081, 2.34752016888707598740907779118, 3.23825086755486375020827984391, 4.25784277552826557251982155070, 5.03806026965038014721162628365, 5.55848905589213998501720184887, 6.36261389471308870138602900888, 6.49786531494913645533761715591, 7.12440344899974255905087582496, 7.896723878259049090482553992495, 8.168656713206124872277685368931, 8.533161498536631865986006945074, 8.603539619290756001226038948684
