| L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 9-s − 6·11-s − 4·14-s + 5·16-s + 2·18-s − 12·22-s + 8·23-s − 9·25-s − 6·28-s + 6·32-s + 3·36-s − 4·37-s − 12·43-s − 18·44-s + 16·46-s − 3·49-s − 18·50-s + 28·53-s − 8·56-s − 2·63-s + 7·64-s − 14·67-s − 6·71-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 1/3·9-s − 1.80·11-s − 1.06·14-s + 5/4·16-s + 0.471·18-s − 2.55·22-s + 1.66·23-s − 9/5·25-s − 1.13·28-s + 1.06·32-s + 1/2·36-s − 0.657·37-s − 1.82·43-s − 2.71·44-s + 2.35·46-s − 3/7·49-s − 2.54·50-s + 3.84·53-s − 1.06·56-s − 0.251·63-s + 7/8·64-s − 1.71·67-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.33319367325295365011876797445, −7.29244342912013911289423114500, −6.68140606826707863203316289977, −6.21449897841157477524696518952, −5.83067800883127320255072941101, −5.30669754150143302157492662003, −5.03620516471027669405867234682, −4.69595647820141900437338182005, −3.91786012964589383634245254650, −3.54463186962642716865631176024, −3.18440748862651463164388008513, −2.34546937125578111204655036909, −2.34067320256225193072738795871, −1.24435229564058784027526283647, 0,
1.24435229564058784027526283647, 2.34067320256225193072738795871, 2.34546937125578111204655036909, 3.18440748862651463164388008513, 3.54463186962642716865631176024, 3.91786012964589383634245254650, 4.69595647820141900437338182005, 5.03620516471027669405867234682, 5.30669754150143302157492662003, 5.83067800883127320255072941101, 6.21449897841157477524696518952, 6.68140606826707863203316289977, 7.29244342912013911289423114500, 7.33319367325295365011876797445
