| L(s) = 1 | − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 2·7-s − 3·8-s + 2·9-s + 3·10-s − 5·11-s − 2·12-s + 3·13-s + 2·14-s + 6·15-s + 16-s + 17-s − 2·18-s − 3·20-s + 4·21-s + 5·22-s − 4·23-s + 6·24-s − 2·25-s − 3·26-s − 6·27-s − 2·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 2/3·9-s + 0.948·10-s − 1.50·11-s − 0.577·12-s + 0.832·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s − 0.670·20-s + 0.872·21-s + 1.06·22-s − 0.834·23-s + 1.22·24-s − 2/5·25-s − 0.588·26-s − 1.15·27-s − 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100261 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100261 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 14323 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 140 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 44 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 79 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 85 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 80 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−14.7992999298, −13.9384097977, −13.4859603485, −12.9754611624, −12.5493323942, −12.2920649272, −11.6239142954, −11.3551228522, −11.1004811691, −10.6755118822, −9.92828796821, −9.71686010661, −9.17872623698, −8.43721878248, −8.04662345428, −7.59627552193, −7.26188269229, −6.29311804705, −6.21954518509, −5.58750527351, −5.00804612579, −4.14559391193, −3.55260908136, −2.99167965785, −1.85855710913, 0, 0,
1.85855710913, 2.99167965785, 3.55260908136, 4.14559391193, 5.00804612579, 5.58750527351, 6.21954518509, 6.29311804705, 7.26188269229, 7.59627552193, 8.04662345428, 8.43721878248, 9.17872623698, 9.71686010661, 9.92828796821, 10.6755118822, 11.1004811691, 11.3551228522, 11.6239142954, 12.2920649272, 12.5493323942, 12.9754611624, 13.4859603485, 13.9384097977, 14.7992999298
