L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s − 6·9-s + 4·10-s − 4·13-s − 16-s + 2·17-s + 6·18-s + 4·20-s + 2·25-s + 4·26-s + 12·29-s − 5·32-s − 2·34-s + 6·36-s − 4·37-s − 12·40-s − 12·41-s + 24·45-s + 2·49-s − 2·50-s + 4·52-s + 12·53-s − 12·58-s − 20·61-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s − 2·9-s + 1.26·10-s − 1.10·13-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.894·20-s + 2/5·25-s + 0.784·26-s + 2.22·29-s − 0.883·32-s − 0.342·34-s + 36-s − 0.657·37-s − 1.89·40-s − 1.87·41-s + 3.57·45-s + 2/7·49-s − 0.282·50-s + 0.554·52-s + 1.64·53-s − 1.57·58-s − 2.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−11.93623114442209313196485845061, −11.77779497781898747190652445510, −10.71734099889110000433050476633, −10.40227615321498563936322188024, −9.578693167598756855048719447448, −8.695686711870280818326611584019, −8.535338766321817121127070704904, −7.81910395523808377988054564239, −7.50901244752622102880251473780, −6.48235693417520450637241002127, −5.37611913672208711896121353581, −4.74199315541377008016560012773, −3.78911546643884845747068338400, −2.88331668043848910381979864823, 0,
2.88331668043848910381979864823, 3.78911546643884845747068338400, 4.74199315541377008016560012773, 5.37611913672208711896121353581, 6.48235693417520450637241002127, 7.50901244752622102880251473780, 7.81910395523808377988054564239, 8.535338766321817121127070704904, 8.695686711870280818326611584019, 9.578693167598756855048719447448, 10.40227615321498563936322188024, 10.71734099889110000433050476633, 11.77779497781898747190652445510, 11.93623114442209313196485845061