| L(s) = 1 | − 2·7-s − 4·11-s + 3·13-s + 4·17-s − 8·19-s − 4·23-s + 5·25-s + 6·31-s − 10·37-s + 4·41-s − 9·43-s − 10·47-s − 11·49-s − 4·53-s + 14·59-s − 11·61-s + 3·67-s − 14·71-s + 11·73-s + 8·77-s + 79-s + 16·83-s − 14·89-s − 6·91-s − 2·97-s − 10·101-s + 6·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 0.832·13-s + 0.970·17-s − 1.83·19-s − 0.834·23-s + 25-s + 1.07·31-s − 1.64·37-s + 0.624·41-s − 1.37·43-s − 1.45·47-s − 1.57·49-s − 0.549·53-s + 1.82·59-s − 1.40·61-s + 0.366·67-s − 1.66·71-s + 1.28·73-s + 0.911·77-s + 0.112·79-s + 1.75·83-s − 1.48·89-s − 0.628·91-s − 0.203·97-s − 0.995·101-s + 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4188597130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4188597130\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T + 125 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 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
−9.011708647492945420120669559810, −8.518480271794223725325165149378, −8.213496544847131166094314044002, −7.896486269005897439567859813887, −7.81917103414825971954623642986, −6.80695692885730810265991100203, −6.70053956205680381026279992250, −6.50234408314565223826133130662, −6.04492871575234126600456227681, −5.34283757442321885089513625322, −5.33513026917926569138637212403, −4.70758701232997568595353666594, −4.30514794206535342587555458358, −3.66021506325547834898682859232, −3.45691490806085514991615931488, −2.80731273971856373700670131271, −2.57520345403634591764875152523, −1.74217892660147439893482688745, −1.33785728387919940032344961685, −0.20797960963389775676988670256,
0.20797960963389775676988670256, 1.33785728387919940032344961685, 1.74217892660147439893482688745, 2.57520345403634591764875152523, 2.80731273971856373700670131271, 3.45691490806085514991615931488, 3.66021506325547834898682859232, 4.30514794206535342587555458358, 4.70758701232997568595353666594, 5.33513026917926569138637212403, 5.34283757442321885089513625322, 6.04492871575234126600456227681, 6.50234408314565223826133130662, 6.70053956205680381026279992250, 6.80695692885730810265991100203, 7.81917103414825971954623642986, 7.896486269005897439567859813887, 8.213496544847131166094314044002, 8.518480271794223725325165149378, 9.011708647492945420120669559810
