| L(s) = 1 | + 3-s − 2·4-s + 3·7-s − 2·9-s − 2·12-s + 2·13-s − 3·19-s + 3·21-s − 3·25-s − 5·27-s − 6·28-s − 31-s + 4·36-s − 13·37-s + 2·39-s + 16·43-s + 2·49-s − 4·52-s − 3·57-s − 6·63-s + 8·64-s + 21·67-s − 5·73-s − 3·75-s + 6·76-s − 25·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.13·7-s − 2/3·9-s − 0.577·12-s + 0.554·13-s − 0.688·19-s + 0.654·21-s − 3/5·25-s − 0.962·27-s − 1.13·28-s − 0.179·31-s + 2/3·36-s − 2.13·37-s + 0.320·39-s + 2.43·43-s + 2/7·49-s − 0.554·52-s − 0.397·57-s − 0.755·63-s + 64-s + 2.56·67-s − 0.585·73-s − 0.346·75-s + 0.688·76-s − 2.81·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 1021 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.424036887663546570198648840785, −8.085475761546068793572907566637, −7.65501206675206392127566573968, −7.03614175219109605639723880081, −6.57746070916000794931138401285, −5.79084430198346401359893936623, −5.48455889715798123411165206386, −5.06671614397286644063726715115, −4.32592034761407320992470912355, −4.05322618891854994360263710762, −3.53727040837283068920094767150, −2.67733196041197725573620094704, −2.11790588581109296698180584237, −1.31415615737873275655069360860, 0,
1.31415615737873275655069360860, 2.11790588581109296698180584237, 2.67733196041197725573620094704, 3.53727040837283068920094767150, 4.05322618891854994360263710762, 4.32592034761407320992470912355, 5.06671614397286644063726715115, 5.48455889715798123411165206386, 5.79084430198346401359893936623, 6.57746070916000794931138401285, 7.03614175219109605639723880081, 7.65501206675206392127566573968, 8.085475761546068793572907566637, 8.424036887663546570198648840785
