| L(s) = 1 | + 3-s + 9-s + 6·11-s + 2·13-s − 4·23-s + 25-s + 27-s + 6·33-s − 6·37-s + 2·39-s + 16·47-s + 2·49-s + 10·59-s − 4·61-s − 4·69-s − 8·71-s + 75-s + 81-s − 4·83-s + 4·97-s + 6·99-s + 16·107-s − 4·109-s − 6·111-s + 2·117-s + 6·121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.04·33-s − 0.986·37-s + 0.320·39-s + 2.33·47-s + 2/7·49-s + 1.30·59-s − 0.512·61-s − 0.481·69-s − 0.949·71-s + 0.115·75-s + 1/9·81-s − 0.439·83-s + 0.406·97-s + 0.603·99-s + 1.54·107-s − 0.383·109-s − 0.569·111-s + 0.184·117-s + 6/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.385748785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385748785\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.005505796026119010491748277991, −8.765919343683970342956211559566, −8.480488222809937884936848210745, −7.68176154632423190480989975701, −7.29394289492785598831313393348, −6.80038917381843720888026626138, −6.25507060472040705944199542486, −5.86876509264739336819843975730, −5.17109307059675204415364773313, −4.36823384763118061986049932553, −3.91529554303995695655690167573, −3.57907540466922684480604372568, −2.68862193718230960330629920498, −1.87876980403692906336229093676, −1.09473876352937500245833237569,
1.09473876352937500245833237569, 1.87876980403692906336229093676, 2.68862193718230960330629920498, 3.57907540466922684480604372568, 3.91529554303995695655690167573, 4.36823384763118061986049932553, 5.17109307059675204415364773313, 5.86876509264739336819843975730, 6.25507060472040705944199542486, 6.80038917381843720888026626138, 7.29394289492785598831313393348, 7.68176154632423190480989975701, 8.480488222809937884936848210745, 8.765919343683970342956211559566, 9.005505796026119010491748277991
