| L(s) = 1 | + 2-s − 2·3-s − 2·4-s − 3·5-s − 2·6-s + 7-s − 3·8-s − 3·9-s − 3·10-s + 2·11-s + 4·12-s − 2·13-s + 14-s + 6·15-s + 16-s − 4·17-s − 3·18-s − 10·19-s + 6·20-s − 2·21-s + 2·22-s − 2·23-s + 6·24-s − 2·25-s − 2·26-s + 14·27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 4-s − 1.34·5-s − 0.816·6-s + 0.377·7-s − 1.06·8-s − 9-s − 0.948·10-s + 0.603·11-s + 1.15·12-s − 0.554·13-s + 0.267·14-s + 1.54·15-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 2.29·19-s + 1.34·20-s − 0.436·21-s + 0.426·22-s − 0.417·23-s + 1.22·24-s − 2/5·25-s − 0.392·26-s + 2.69·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 9 T + 83 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 23 T + 297 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−11.56426593197010668947299479173, −11.05891393307674927129326086928, −10.47412010902357538531671035869, −10.23231236727852735158187755764, −9.091535230729964662820616315035, −9.072438898250098431588924716074, −8.308670724881931507901874883122, −8.265319807331788902528017453824, −7.53375436025889053870889076761, −6.71722187275342871662693756837, −6.12483968027025517708425791360, −6.08177235341098556091878867939, −4.98762802351003999433345023780, −4.85743213623708093591136946478, −4.34228849486437772089495557350, −3.83340761680378082441701435865, −3.16319960542823909784227014131, −2.07327473077312776756495047181, 0, 0,
2.07327473077312776756495047181, 3.16319960542823909784227014131, 3.83340761680378082441701435865, 4.34228849486437772089495557350, 4.85743213623708093591136946478, 4.98762802351003999433345023780, 6.08177235341098556091878867939, 6.12483968027025517708425791360, 6.71722187275342871662693756837, 7.53375436025889053870889076761, 8.265319807331788902528017453824, 8.308670724881931507901874883122, 9.072438898250098431588924716074, 9.091535230729964662820616315035, 10.23231236727852735158187755764, 10.47412010902357538531671035869, 11.05891393307674927129326086928, 11.56426593197010668947299479173
