| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s + 4·8-s + 3·9-s − 4·10-s − 2·11-s − 6·12-s + 4·15-s + 5·16-s − 6·17-s + 6·18-s − 2·19-s − 6·20-s − 4·22-s + 4·23-s − 8·24-s − 2·25-s − 4·27-s + 8·30-s − 10·31-s + 6·32-s + 4·33-s − 12·34-s + 9·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s − 1.73·12-s + 1.03·15-s + 5/4·16-s − 1.45·17-s + 1.41·18-s − 0.458·19-s − 1.34·20-s − 0.852·22-s + 0.834·23-s − 1.63·24-s − 2/5·25-s − 0.769·27-s + 1.46·30-s − 1.79·31-s + 1.06·32-s + 0.696·33-s − 2.05·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10458756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 162 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.308395201887522467306379368116, −7.937319973966220905190776043721, −7.45554471439769903633781626731, −7.25495299417794870485325880886, −6.87453239831790004284162762187, −6.51425790032642295735257257158, −6.06269675782315864237222366731, −5.86755441476510750574776400246, −5.20488699111769931025085260878, −5.08140986852376064045951094602, −4.48584054206085685239316898852, −4.48494736581129919926470525921, −3.74614230975455923025774579111, −3.57322553777284998809167820847, −2.96250317694907553878966878241, −2.43139618248874823913260659253, −1.78668960341985262097848069562, −1.39274818428326106592629821662, 0, 0,
1.39274818428326106592629821662, 1.78668960341985262097848069562, 2.43139618248874823913260659253, 2.96250317694907553878966878241, 3.57322553777284998809167820847, 3.74614230975455923025774579111, 4.48494736581129919926470525921, 4.48584054206085685239316898852, 5.08140986852376064045951094602, 5.20488699111769931025085260878, 5.86755441476510750574776400246, 6.06269675782315864237222366731, 6.51425790032642295735257257158, 6.87453239831790004284162762187, 7.25495299417794870485325880886, 7.45554471439769903633781626731, 7.937319973966220905190776043721, 8.308395201887522467306379368116
