| L(s) = 1 | − 2-s − 4-s − 5-s + 8-s − 2·9-s + 10-s − 4·13-s − 16-s + 3·17-s + 2·18-s + 9·19-s + 20-s − 6·23-s + 2·25-s + 4·26-s − 3·27-s − 5·29-s + 2·31-s + 5·32-s − 3·34-s + 2·36-s − 5·37-s − 9·38-s − 40-s + 13·41-s − 43-s + 2·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.353·8-s − 2/3·9-s + 0.316·10-s − 1.10·13-s − 1/4·16-s + 0.727·17-s + 0.471·18-s + 2.06·19-s + 0.223·20-s − 1.25·23-s + 2/5·25-s + 0.784·26-s − 0.577·27-s − 0.928·29-s + 0.359·31-s + 0.883·32-s − 0.514·34-s + 1/3·36-s − 0.821·37-s − 1.45·38-s − 0.158·40-s + 2.03·41-s − 0.152·43-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 382989 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 382989 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 127663 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 24 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 21 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 119 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + T - 37 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 68 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 31 T + 403 T^{2} - 31 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 93 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 105 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−13.1213387805, −12.5669248194, −12.0792607371, −11.8629926811, −11.4632742732, −11.1357187395, −10.4388590298, −10.0147830975, −9.64428527614, −9.29353826170, −9.15388357882, −8.23477797759, −8.12457655023, −7.61732051223, −7.38427273967, −6.64389392602, −6.12728946181, −5.49595826742, −5.09083061289, −4.73760099972, −3.81373587730, −3.51594821949, −2.77814057919, −2.07758576054, −0.947805959443, 0,
0.947805959443, 2.07758576054, 2.77814057919, 3.51594821949, 3.81373587730, 4.73760099972, 5.09083061289, 5.49595826742, 6.12728946181, 6.64389392602, 7.38427273967, 7.61732051223, 8.12457655023, 8.23477797759, 9.15388357882, 9.29353826170, 9.64428527614, 10.0147830975, 10.4388590298, 11.1357187395, 11.4632742732, 11.8629926811, 12.0792607371, 12.5669248194, 13.1213387805
