| L(s) = 1 | − 2·2-s − 3-s + 2·6-s − 3·7-s + 4·8-s + 2·9-s − 5·11-s − 5·13-s + 6·14-s − 4·16-s − 4·17-s − 4·18-s − 3·19-s + 3·21-s + 10·22-s + 3·23-s − 4·24-s − 2·25-s + 10·26-s − 6·27-s + 4·29-s − 31-s + 5·33-s + 8·34-s + 37-s + 6·38-s + 5·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 0.816·6-s − 1.13·7-s + 1.41·8-s + 2/3·9-s − 1.50·11-s − 1.38·13-s + 1.60·14-s − 16-s − 0.970·17-s − 0.942·18-s − 0.688·19-s + 0.654·21-s + 2.13·22-s + 0.625·23-s − 0.816·24-s − 2/5·25-s + 1.96·26-s − 1.15·27-s + 0.742·29-s − 0.179·31-s + 0.870·33-s + 1.37·34-s + 0.164·37-s + 0.973·38-s + 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118187 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118187 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 1619 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 52 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 19 T + 170 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14 T + 134 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T - 38 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T - 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T - 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 119 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.3852482393, −13.6678846463, −13.2963999253, −12.9830650779, −12.6914611052, −12.2949764512, −11.5247144531, −11.0647104519, −10.6162860635, −10.2324510737, −9.72281351992, −9.52617316556, −9.28176827115, −8.40807569090, −8.05738040277, −7.72964425705, −7.03499270782, −6.62523308073, −6.09499412306, −5.25428853352, −4.70238906190, −4.51163400725, −3.37028744837, −2.66757037739, −1.71141061986, 0, 0,
1.71141061986, 2.66757037739, 3.37028744837, 4.51163400725, 4.70238906190, 5.25428853352, 6.09499412306, 6.62523308073, 7.03499270782, 7.72964425705, 8.05738040277, 8.40807569090, 9.28176827115, 9.52617316556, 9.72281351992, 10.2324510737, 10.6162860635, 11.0647104519, 11.5247144531, 12.2949764512, 12.6914611052, 12.9830650779, 13.2963999253, 13.6678846463, 14.3852482393
