| L(s) = 1 | − 3·2-s − 3-s + 4·4-s + 3·6-s − 3·7-s − 3·8-s − 3·9-s − 11-s − 4·12-s − 13-s + 9·14-s + 3·16-s − 4·17-s + 9·18-s − 3·19-s + 3·21-s + 3·22-s − 4·23-s + 3·24-s + 25-s + 3·26-s + 4·27-s − 12·28-s − 31-s − 6·32-s + 33-s + 12·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 2·4-s + 1.22·6-s − 1.13·7-s − 1.06·8-s − 9-s − 0.301·11-s − 1.15·12-s − 0.277·13-s + 2.40·14-s + 3/4·16-s − 0.970·17-s + 2.12·18-s − 0.688·19-s + 0.654·21-s + 0.639·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.588·26-s + 0.769·27-s − 2.26·28-s − 0.179·31-s − 1.06·32-s + 0.174·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100121 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 14303 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 170 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 32 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 55 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 14 T + 105 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 68 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 48 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 112 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 171 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 25 T + 345 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
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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
−14.7706469544, −13.9370107649, −13.5431774477, −13.2077106818, −12.5423592213, −12.1030651912, −11.7625322305, −11.1799773110, −10.7788794290, −10.2233909024, −10.0445417434, −9.60053715731, −8.96930180070, −8.61923043099, −8.44799012975, −7.79876663782, −7.17687310436, −6.61381146466, −6.29480109232, −5.63515366861, −5.08595632053, −4.17698504246, −3.29562345941, −2.64889963602, −1.65922345034, 0, 0,
1.65922345034, 2.64889963602, 3.29562345941, 4.17698504246, 5.08595632053, 5.63515366861, 6.29480109232, 6.61381146466, 7.17687310436, 7.79876663782, 8.44799012975, 8.61923043099, 8.96930180070, 9.60053715731, 10.0445417434, 10.2233909024, 10.7788794290, 11.1799773110, 11.7625322305, 12.1030651912, 12.5423592213, 13.2077106818, 13.5431774477, 13.9370107649, 14.7706469544
