| L(s) = 1 | − 2-s − 3-s − 2·4-s + 5-s + 6-s − 4·7-s + 3·8-s − 2·9-s − 10-s − 2·11-s + 2·12-s − 2·13-s + 4·14-s − 15-s + 16-s − 3·17-s + 2·18-s − 9·19-s − 2·20-s + 4·21-s + 2·22-s − 23-s − 3·24-s − 4·25-s + 2·26-s + 5·27-s + 8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 2.06·19-s − 0.447·20-s + 0.872·21-s + 0.426·22-s − 0.208·23-s − 0.612·24-s − 4/5·25-s + 0.392·26-s + 0.962·27-s + 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100089 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 337 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 67 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 81 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 110 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 95 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 183 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 190 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 233 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−14.5333877522, −13.8835759980, −13.5768764663, −13.0288312747, −12.7533038172, −12.6453731078, −11.7871657601, −11.3706604964, −10.6881648609, −10.3833699495, −10.0394936852, −9.53469632888, −9.04965380686, −8.75719609829, −8.36251358943, −7.67533012864, −6.95290896824, −6.58349519904, −5.94672683229, −5.53635827534, −4.96573808986, −4.20132187211, −3.72249758303, −2.74332991356, −2.01770084709, 0, 0,
2.01770084709, 2.74332991356, 3.72249758303, 4.20132187211, 4.96573808986, 5.53635827534, 5.94672683229, 6.58349519904, 6.95290896824, 7.67533012864, 8.36251358943, 8.75719609829, 9.04965380686, 9.53469632888, 10.0394936852, 10.3833699495, 10.6881648609, 11.3706604964, 11.7871657601, 12.6453731078, 12.7533038172, 13.0288312747, 13.5768764663, 13.8835759980, 14.5333877522
