| L(s) = 1 | + 2-s − 3-s + 4-s − 5·5-s − 6-s + 8-s + 9-s − 5·10-s − 12-s + 5·15-s + 16-s + 18-s + 4·19-s − 5·20-s − 12·23-s − 24-s + 16·25-s − 27-s − 29-s + 5·30-s + 32-s + 36-s + 4·38-s − 5·40-s + 14·43-s − 5·45-s − 12·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 2.23·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.58·10-s − 0.288·12-s + 1.29·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 1.11·20-s − 2.50·23-s − 0.204·24-s + 16/5·25-s − 0.192·27-s − 0.185·29-s + 0.912·30-s + 0.176·32-s + 1/6·36-s + 0.648·38-s − 0.790·40-s + 2.13·43-s − 0.745·45-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
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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
−8.131799507428394894547507053540, −7.59681338232866011965939794997, −7.37679753750141448551285770288, −7.19407231691601327950318155180, −6.29702866154259534899741164402, −5.95758522002800123945372546882, −5.51565974595561619309654634071, −4.77515668403916234972415738313, −4.32409356733915756479303575476, −4.05369308597619691143636499291, −3.56728205164927763031414818414, −2.98838478225608794273623739170, −2.17390276414206357580549248276, −1.03603055340586359314888043406, 0,
1.03603055340586359314888043406, 2.17390276414206357580549248276, 2.98838478225608794273623739170, 3.56728205164927763031414818414, 4.05369308597619691143636499291, 4.32409356733915756479303575476, 4.77515668403916234972415738313, 5.51565974595561619309654634071, 5.95758522002800123945372546882, 6.29702866154259534899741164402, 7.19407231691601327950318155180, 7.37679753750141448551285770288, 7.59681338232866011965939794997, 8.131799507428394894547507053540
