L(s) = 1 | + 3-s − 3·4-s + 2·7-s + 9-s − 3·12-s − 12·13-s + 5·16-s − 16·19-s + 2·21-s + 25-s + 27-s − 6·28-s + 8·31-s − 3·36-s − 4·37-s − 12·39-s + 8·43-s + 5·48-s + 3·49-s + 36·52-s − 16·57-s − 4·61-s + 2·63-s − 3·64-s + 8·67-s − 4·73-s + 75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 3/2·4-s + 0.755·7-s + 1/3·9-s − 0.866·12-s − 3.32·13-s + 5/4·16-s − 3.67·19-s + 0.436·21-s + 1/5·25-s + 0.192·27-s − 1.13·28-s + 1.43·31-s − 1/2·36-s − 0.657·37-s − 1.92·39-s + 1.21·43-s + 0.721·48-s + 3/7·49-s + 4.99·52-s − 2.11·57-s − 0.512·61-s + 0.251·63-s − 3/8·64-s + 0.977·67-s − 0.468·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−10.15928312968820765258481052050, −9.386500711260631787757553076630, −9.326498061922569555987691090592, −8.447489066361171687346291758504, −8.261177977090796325600765542566, −7.73831918709431562684545235915, −6.97944382340517249810347443111, −6.48451110526696282246961711325, −5.37573342552736399524296425305, −4.87484917945516723546545158294, −4.33309058948755389799390879771, −4.17084275561163201389021211529, −2.63294756479994443235170715895, −2.17599002568151719086793905114, 0,
2.17599002568151719086793905114, 2.63294756479994443235170715895, 4.17084275561163201389021211529, 4.33309058948755389799390879771, 4.87484917945516723546545158294, 5.37573342552736399524296425305, 6.48451110526696282246961711325, 6.97944382340517249810347443111, 7.73831918709431562684545235915, 8.261177977090796325600765542566, 8.447489066361171687346291758504, 9.326498061922569555987691090592, 9.386500711260631787757553076630, 10.15928312968820765258481052050