L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 4·14-s + 16-s − 2·17-s − 6·23-s − 25-s + 4·28-s + 2·31-s + 32-s − 2·34-s + 16·41-s − 6·46-s + 12·47-s + 9·49-s − 50-s + 4·56-s + 2·62-s + 64-s − 2·68-s + 2·71-s + 2·73-s + 6·79-s − 9·81-s + 16·82-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.06·14-s + 1/4·16-s − 0.485·17-s − 1.25·23-s − 1/5·25-s + 0.755·28-s + 0.359·31-s + 0.176·32-s − 0.342·34-s + 2.49·41-s − 0.884·46-s + 1.75·47-s + 9/7·49-s − 0.141·50-s + 0.534·56-s + 0.254·62-s + 1/8·64-s − 0.242·68-s + 0.237·71-s + 0.234·73-s + 0.675·79-s − 81-s + 1.76·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.063370518\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.063370518\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 100 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.048389153069890758028969391922, −8.859992151483496950116946934550, −8.094228257415729708827960994117, −7.76435957436269231351177197973, −7.45769195765569686691230186424, −6.76974376490135645948244733818, −6.08487267499654193596351061171, −5.77842780177725906519637613672, −5.16343057058219948774063457296, −4.55162189574507980507465442060, −4.20578495356452115826261311677, −3.64637951009165279880210659698, −2.51855059844139618506209602713, −2.18039551376199327715807383460, −1.16055179583190924813761668775,
1.16055179583190924813761668775, 2.18039551376199327715807383460, 2.51855059844139618506209602713, 3.64637951009165279880210659698, 4.20578495356452115826261311677, 4.55162189574507980507465442060, 5.16343057058219948774063457296, 5.77842780177725906519637613672, 6.08487267499654193596351061171, 6.76974376490135645948244733818, 7.45769195765569686691230186424, 7.76435957436269231351177197973, 8.094228257415729708827960994117, 8.859992151483496950116946934550, 9.048389153069890758028969391922