| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·9-s − 2·10-s − 6·13-s − 16-s + 2·18-s − 2·20-s − 25-s + 6·26-s − 12·29-s − 5·32-s + 2·36-s + 12·37-s + 6·40-s − 4·45-s − 14·49-s + 50-s + 6·52-s + 12·58-s − 12·61-s + 7·64-s − 12·65-s − 6·72-s − 12·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 2/3·9-s − 0.632·10-s − 1.66·13-s − 1/4·16-s + 0.471·18-s − 0.447·20-s − 1/5·25-s + 1.17·26-s − 2.22·29-s − 0.883·32-s + 1/3·36-s + 1.97·37-s + 0.948·40-s − 0.596·45-s − 2·49-s + 0.141·50-s + 0.832·52-s + 1.57·58-s − 1.53·61-s + 7/8·64-s − 1.48·65-s − 0.707·72-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−9.675507463026625542878847598321, −9.148341510089643709064217437508, −8.889860752789099165267891779907, −7.983338143927186391141727043542, −7.70109681955809044075586522751, −7.31593835346156771513900334891, −6.45386612590213730965224731681, −5.86961066422327844959030933926, −5.38727946505914976028322322688, −4.78576617982490866904832274281, −4.23173266261608360047454002997, −3.25122517033836649296126383801, −2.40077649629253122060261729388, −1.65402705683659248810196888504, 0,
1.65402705683659248810196888504, 2.40077649629253122060261729388, 3.25122517033836649296126383801, 4.23173266261608360047454002997, 4.78576617982490866904832274281, 5.38727946505914976028322322688, 5.86961066422327844959030933926, 6.45386612590213730965224731681, 7.31593835346156771513900334891, 7.70109681955809044075586522751, 7.983338143927186391141727043542, 8.889860752789099165267891779907, 9.148341510089643709064217437508, 9.675507463026625542878847598321
