| L(s) = 1 | + 3-s + 4-s − 2·7-s + 9-s + 12-s + 2·13-s + 16-s − 8·19-s − 2·21-s − 6·25-s + 27-s − 2·28-s + 36-s − 4·37-s + 2·39-s + 8·43-s + 48-s + 3·49-s + 2·52-s − 8·57-s − 20·61-s − 2·63-s + 64-s + 8·67-s − 12·73-s − 6·75-s − 8·76-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 0.755·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s − 0.377·28-s + 1/6·36-s − 0.657·37-s + 0.320·39-s + 1.21·43-s + 0.144·48-s + 3/7·49-s + 0.277·52-s − 1.05·57-s − 2.56·61-s − 0.251·63-s + 1/8·64-s + 0.977·67-s − 1.40·73-s − 0.692·75-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.919520600930853256251460201426, −7.60950019354364822141375551305, −7.05482338122635591122480401032, −6.62410620605336828325420319307, −6.18971912035243321530580658857, −5.93037987998780819768001729788, −5.36734378010452113216798273238, −4.44015518719984939594103992774, −4.32571873646081315893153040604, −3.48844535790283312124632987251, −3.34291279159901837009808602924, −2.39519789676916156425539342752, −2.15328177975657163080119509125, −1.29507569141715195434836244797, 0,
1.29507569141715195434836244797, 2.15328177975657163080119509125, 2.39519789676916156425539342752, 3.34291279159901837009808602924, 3.48844535790283312124632987251, 4.32571873646081315893153040604, 4.44015518719984939594103992774, 5.36734378010452113216798273238, 5.93037987998780819768001729788, 6.18971912035243321530580658857, 6.62410620605336828325420319307, 7.05482338122635591122480401032, 7.60950019354364822141375551305, 7.919520600930853256251460201426
