| L(s) = 1 | − 2·5-s + 2·7-s + 2·11-s + 13-s + 12·17-s − 12·19-s − 8·23-s + 5·25-s − 2·29-s − 10·31-s − 4·35-s − 12·37-s + 6·41-s − 4·43-s + 2·47-s + 7·49-s + 12·53-s − 4·55-s + 10·59-s + 2·61-s − 2·65-s − 10·67-s + 20·71-s + 4·73-s + 4·77-s + 4·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s + 0.603·11-s + 0.277·13-s + 2.91·17-s − 2.75·19-s − 1.66·23-s + 25-s − 0.371·29-s − 1.79·31-s − 0.676·35-s − 1.97·37-s + 0.937·41-s − 0.609·43-s + 0.291·47-s + 49-s + 1.64·53-s − 0.539·55-s + 1.30·59-s + 0.256·61-s − 0.248·65-s − 1.22·67-s + 2.37·71-s + 0.468·73-s + 0.455·77-s + 0.450·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17740944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.270829966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.270829966\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.624265590424849915600391747664, −8.128286327613359879013789263651, −7.980767723666915448596498309910, −7.55089957796345854712786447494, −7.23597426317553315411605002760, −6.70444702205413018812873600587, −6.55802763671425118560280122004, −5.90201570534229738204400098804, −5.59472594954575488756539201543, −5.26480494130717158267404582058, −4.97671880569526842381040977291, −4.16349127212245177178904084549, −3.90550110446797080852529747824, −3.76899013665345692418308127054, −3.50573402356748327702214774980, −2.44293332472713963989103048550, −2.37240852920154976805993248198, −1.50184267658733535969875812488, −1.29819833165106485558294966642, −0.33041763291423011087851613387,
0.33041763291423011087851613387, 1.29819833165106485558294966642, 1.50184267658733535969875812488, 2.37240852920154976805993248198, 2.44293332472713963989103048550, 3.50573402356748327702214774980, 3.76899013665345692418308127054, 3.90550110446797080852529747824, 4.16349127212245177178904084549, 4.97671880569526842381040977291, 5.26480494130717158267404582058, 5.59472594954575488756539201543, 5.90201570534229738204400098804, 6.55802763671425118560280122004, 6.70444702205413018812873600587, 7.23597426317553315411605002760, 7.55089957796345854712786447494, 7.980767723666915448596498309910, 8.128286327613359879013789263651, 8.624265590424849915600391747664
