| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s + 3·9-s + 8·10-s − 4·12-s + 4·13-s + 8·15-s − 4·16-s − 6·18-s − 8·20-s + 11·25-s − 8·26-s − 4·27-s − 16·30-s − 8·31-s + 8·32-s + 6·36-s − 12·37-s − 8·39-s − 20·43-s − 12·45-s + 8·48-s − 49-s − 22·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 9-s + 2.52·10-s − 1.15·12-s + 1.10·13-s + 2.06·15-s − 16-s − 1.41·18-s − 1.78·20-s + 11/5·25-s − 1.56·26-s − 0.769·27-s − 2.92·30-s − 1.43·31-s + 1.41·32-s + 36-s − 1.97·37-s − 1.28·39-s − 3.04·43-s − 1.78·45-s + 1.15·48-s − 1/7·49-s − 3.11·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.933622423645576485119565175719, −9.734943247681086969613932545714, −9.004404498362401074274100081351, −8.651777371744057173275049665288, −8.275549062173679088523793607564, −8.062441809345203619964575774025, −7.36785276919967290597936632437, −7.10135286658060362424950938518, −6.79315568196857936322539010217, −6.26781392186522521072674815450, −5.66509429335177534367185896233, −4.99064124873252736738842142605, −4.67346177889569119987403938465, −4.10000995042128240493500097903, −3.45174096519215580512949610758, −3.15260796033033971278292888267, −1.65603733781226436850893871696, −1.40769865256966206039999480012, 0, 0,
1.40769865256966206039999480012, 1.65603733781226436850893871696, 3.15260796033033971278292888267, 3.45174096519215580512949610758, 4.10000995042128240493500097903, 4.67346177889569119987403938465, 4.99064124873252736738842142605, 5.66509429335177534367185896233, 6.26781392186522521072674815450, 6.79315568196857936322539010217, 7.10135286658060362424950938518, 7.36785276919967290597936632437, 8.062441809345203619964575774025, 8.275549062173679088523793607564, 8.651777371744057173275049665288, 9.004404498362401074274100081351, 9.734943247681086969613932545714, 9.933622423645576485119565175719
