L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 3·9-s − 2·10-s − 2·13-s + 16-s − 6·17-s + 3·18-s − 2·20-s − 7·25-s − 2·26-s + 4·29-s + 32-s − 6·34-s + 3·36-s + 6·37-s − 2·40-s − 6·45-s − 13·49-s − 7·50-s − 2·52-s + 24·53-s + 4·58-s − 16·61-s + 64-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 9-s − 0.632·10-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.447·20-s − 7/5·25-s − 0.392·26-s + 0.742·29-s + 0.176·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.316·40-s − 0.894·45-s − 1.85·49-s − 0.989·50-s − 0.277·52-s + 3.29·53-s + 0.525·58-s − 2.04·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.120257005\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.120257005\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−12.14995362405968847306495338636, −11.49712673239204411601833482255, −11.44138555694141255918714539737, −10.28025762974353066563394220777, −10.16479571855116242915645828857, −9.175534225967753404519516979495, −8.542273196799616534200427297006, −7.53566579558648020530867420842, −7.45985560667588965978165906144, −6.53148501748885385532746785562, −5.88908450915949561478136129218, −4.61153453491182141362175201622, −4.43437896316507372499242030963, −3.47529908355398035375426875224, −2.17545984222807868215828295765,
2.17545984222807868215828295765, 3.47529908355398035375426875224, 4.43437896316507372499242030963, 4.61153453491182141362175201622, 5.88908450915949561478136129218, 6.53148501748885385532746785562, 7.45985560667588965978165906144, 7.53566579558648020530867420842, 8.542273196799616534200427297006, 9.175534225967753404519516979495, 10.16479571855116242915645828857, 10.28025762974353066563394220777, 11.44138555694141255918714539737, 11.49712673239204411601833482255, 12.14995362405968847306495338636