| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s + 3·9-s − 2·10-s + 4·11-s + 13-s + 16-s − 6·17-s − 3·18-s − 12·19-s + 2·20-s − 4·22-s − 8·23-s − 7·25-s − 26-s − 32-s + 6·34-s + 3·36-s − 6·37-s + 12·38-s − 2·40-s + 4·44-s + 6·45-s + 8·46-s − 13·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s + 9-s − 0.632·10-s + 1.20·11-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 2.75·19-s + 0.447·20-s − 0.852·22-s − 1.66·23-s − 7/5·25-s − 0.196·26-s − 0.176·32-s + 1.02·34-s + 1/2·36-s − 0.986·37-s + 1.94·38-s − 0.316·40-s + 0.603·44-s + 0.894·45-s + 1.17·46-s − 1.85·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281216 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 13 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.567359956354704438407736576121, −8.310061634639968993764403624165, −7.938211248845934878732993890823, −6.92880645805086748841087984321, −6.79038227196567694631715258566, −6.40941752007862407532170431759, −5.98494658430119741237026147495, −5.41501572629242590776107371268, −4.42326331313024154570777237165, −4.09964629322041262514106195672, −3.79033676077549186455122166647, −2.47488980077832639227670038418, −1.82604108038319412812644968804, −1.74404149014246056815546005306, 0,
1.74404149014246056815546005306, 1.82604108038319412812644968804, 2.47488980077832639227670038418, 3.79033676077549186455122166647, 4.09964629322041262514106195672, 4.42326331313024154570777237165, 5.41501572629242590776107371268, 5.98494658430119741237026147495, 6.40941752007862407532170431759, 6.79038227196567694631715258566, 6.92880645805086748841087984321, 7.938211248845934878732993890823, 8.310061634639968993764403624165, 8.567359956354704438407736576121
