| L(s) = 1 | + 3·7-s − 3·9-s + 5·13-s − 4·16-s − 7·19-s + 3·25-s + 3·31-s − 11·37-s − 13·43-s + 2·49-s + 61-s − 9·63-s − 2·67-s + 8·73-s + 16·79-s + 9·81-s + 15·91-s + 14·97-s − 5·103-s − 4·109-s − 12·112-s − 15·117-s − 14·121-s + 127-s + 131-s − 21·133-s + 137-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 9-s + 1.38·13-s − 16-s − 1.60·19-s + 3/5·25-s + 0.538·31-s − 1.80·37-s − 1.98·43-s + 2/7·49-s + 0.128·61-s − 1.13·63-s − 0.244·67-s + 0.936·73-s + 1.80·79-s + 81-s + 1.57·91-s + 1.42·97-s − 0.492·103-s − 0.383·109-s − 1.13·112-s − 1.38·117-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s − 1.82·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 10 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 87 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−8.075717065117129420870188250669, −7.73186038376666027943616222236, −6.87414542851740960349063609141, −6.57054264109067322228328268947, −6.36111227397660405522879078329, −5.66085148546612320471822587520, −5.17243991005517407695694746982, −4.80302011498473274907205468492, −4.34409242161693997145529149495, −3.61892957759330578167609241579, −3.31707408507968765611063747160, −2.37094415869088425886604487461, −1.99575673943762569248908241299, −1.23297938934624919393394195800, 0,
1.23297938934624919393394195800, 1.99575673943762569248908241299, 2.37094415869088425886604487461, 3.31707408507968765611063747160, 3.61892957759330578167609241579, 4.34409242161693997145529149495, 4.80302011498473274907205468492, 5.17243991005517407695694746982, 5.66085148546612320471822587520, 6.36111227397660405522879078329, 6.57054264109067322228328268947, 6.87414542851740960349063609141, 7.73186038376666027943616222236, 8.075717065117129420870188250669
