| L(s) = 1 | − 2·3-s + 4-s + 9-s − 2·12-s + 2·13-s − 3·16-s − 14·17-s − 5·23-s − 6·25-s + 4·27-s + 36-s − 4·39-s + 10·43-s + 6·48-s − 8·49-s + 28·51-s + 2·52-s + 4·53-s − 7·64-s − 14·68-s + 10·69-s + 12·75-s + 14·79-s − 11·81-s − 5·92-s − 6·100-s − 4·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 1/3·9-s − 0.577·12-s + 0.554·13-s − 3/4·16-s − 3.39·17-s − 1.04·23-s − 6/5·25-s + 0.769·27-s + 1/6·36-s − 0.640·39-s + 1.52·43-s + 0.866·48-s − 8/7·49-s + 3.92·51-s + 0.277·52-s + 0.549·53-s − 7/8·64-s − 1.69·68-s + 1.20·69-s + 1.38·75-s + 1.57·79-s − 1.22·81-s − 0.521·92-s − 3/5·100-s − 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34983 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34983 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−10.41698324386048359017234494898, −9.563481542645849036174752611036, −9.145124376923883484631439073123, −8.586880218166972869237113185503, −8.035993233501511012010807107370, −7.23221019331411544656851850574, −6.61119708730928096552204908189, −6.39170937744355846054941209625, −5.88601784976269795291596213330, −5.09244860106418703651088251610, −4.35718027343160538798056817090, −4.00704820309228774486155208975, −2.57994955081099631330836439386, −1.95606224114920306525269285272, 0,
1.95606224114920306525269285272, 2.57994955081099631330836439386, 4.00704820309228774486155208975, 4.35718027343160538798056817090, 5.09244860106418703651088251610, 5.88601784976269795291596213330, 6.39170937744355846054941209625, 6.61119708730928096552204908189, 7.23221019331411544656851850574, 8.035993233501511012010807107370, 8.586880218166972869237113185503, 9.145124376923883484631439073123, 9.563481542645849036174752611036, 10.41698324386048359017234494898
