| L(s) = 1 | + 3-s − 2·4-s − 2·5-s + 3·7-s + 9-s − 7·11-s − 2·12-s + 8·13-s − 2·15-s − 7·17-s − 6·19-s + 4·20-s + 3·21-s + 2·25-s + 4·27-s − 6·28-s + 9·29-s − 2·31-s − 7·33-s − 6·35-s − 2·36-s + 12·37-s + 8·39-s − 10·41-s + 14·44-s − 2·45-s − 2·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.894·5-s + 1.13·7-s + 1/3·9-s − 2.11·11-s − 0.577·12-s + 2.21·13-s − 0.516·15-s − 1.69·17-s − 1.37·19-s + 0.894·20-s + 0.654·21-s + 2/5·25-s + 0.769·27-s − 1.13·28-s + 1.67·29-s − 0.359·31-s − 1.21·33-s − 1.01·35-s − 1/3·36-s + 1.97·37-s + 1.28·39-s − 1.56·41-s + 2.11·44-s − 0.298·45-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134741 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134741 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 134741 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 284 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 9 T + 61 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 7 T + 106 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 21 T + 218 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 94 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 228 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 155 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 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
−13.7230372013, −13.4895105837, −13.3166247335, −12.9329342959, −12.4504578506, −11.6559797429, −11.3947029551, −10.7810438995, −10.6075437914, −10.3009003485, −9.37807156933, −8.72227046055, −8.55092126317, −8.37217925778, −7.94217221413, −7.37353773354, −6.64226280181, −6.14528880248, −5.41416978135, −4.63745474072, −4.41844144882, −4.13735236220, −3.07674748035, −2.54253828814, −1.47574952109, 0,
1.47574952109, 2.54253828814, 3.07674748035, 4.13735236220, 4.41844144882, 4.63745474072, 5.41416978135, 6.14528880248, 6.64226280181, 7.37353773354, 7.94217221413, 8.37217925778, 8.55092126317, 8.72227046055, 9.37807156933, 10.3009003485, 10.6075437914, 10.7810438995, 11.3947029551, 11.6559797429, 12.4504578506, 12.9329342959, 13.3166247335, 13.4895105837, 13.7230372013
