| L(s) = 1 | − 2·2-s + 3·4-s − 5-s − 8·7-s − 4·8-s + 9-s + 2·10-s + 2·13-s + 16·14-s + 5·16-s − 2·18-s − 3·20-s + 25-s − 4·26-s − 24·28-s − 12·29-s − 6·32-s + 8·35-s + 3·36-s + 4·37-s + 4·40-s − 45-s + 34·49-s − 2·50-s + 6·52-s + 32·56-s + 24·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.447·5-s − 3.02·7-s − 1.41·8-s + 1/3·9-s + 0.632·10-s + 0.554·13-s + 4.27·14-s + 5/4·16-s − 0.471·18-s − 0.670·20-s + 1/5·25-s − 0.784·26-s − 4.53·28-s − 2.22·29-s − 1.06·32-s + 1.35·35-s + 1/2·36-s + 0.657·37-s + 0.632·40-s − 0.149·45-s + 34/7·49-s − 0.282·50-s + 0.832·52-s + 4.27·56-s + 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 760500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.941231322257043910223245152113, −7.55397412652055104429359188790, −7.37431102580997813073405467010, −6.66757469343681711452650518079, −6.42217617652666865799421983967, −6.09907154602198419660987368274, −5.69930835701942778538159225157, −4.83449158660366225526812794793, −3.95694186776025806536992768009, −3.39954423361805099487890109587, −3.36585804145949552210873743484, −2.57473182165147345688800624678, −1.84915649406233336762542821373, −0.72861675057256908847367854482, 0,
0.72861675057256908847367854482, 1.84915649406233336762542821373, 2.57473182165147345688800624678, 3.36585804145949552210873743484, 3.39954423361805099487890109587, 3.95694186776025806536992768009, 4.83449158660366225526812794793, 5.69930835701942778538159225157, 6.09907154602198419660987368274, 6.42217617652666865799421983967, 6.66757469343681711452650518079, 7.37431102580997813073405467010, 7.55397412652055104429359188790, 7.941231322257043910223245152113
