| L(s) = 1 | − 3-s − 3·4-s − 2·5-s + 7-s + 9-s + 3·12-s + 2·15-s + 5·16-s − 4·17-s + 6·20-s − 21-s + 3·25-s − 27-s − 3·28-s − 2·35-s − 3·36-s − 4·37-s + 12·41-s + 8·43-s − 2·45-s − 16·47-s − 5·48-s + 49-s + 4·51-s − 8·59-s − 6·60-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 0.866·12-s + 0.516·15-s + 5/4·16-s − 0.970·17-s + 1.34·20-s − 0.218·21-s + 3/5·25-s − 0.192·27-s − 0.566·28-s − 0.338·35-s − 1/2·36-s − 0.657·37-s + 1.87·41-s + 1.21·43-s − 0.298·45-s − 2.33·47-s − 0.721·48-s + 1/7·49-s + 0.560·51-s − 1.04·59-s − 0.774·60-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231525 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.843078054411908501657519136744, −8.261177977090796325600765542566, −7.73831918709431562684545235915, −7.69164818258356184893378968692, −6.67009631001529545506185276841, −6.48451110526696282246961711325, −5.69208011630684437322158261218, −5.01564976002989168468776414543, −4.87484917945516723546545158294, −4.17084275561163201389021211529, −3.94068846921577588199342611811, −3.18167275817234212778601509206, −2.17599002568151719086793905114, −0.982469269817440515290174005672, 0,
0.982469269817440515290174005672, 2.17599002568151719086793905114, 3.18167275817234212778601509206, 3.94068846921577588199342611811, 4.17084275561163201389021211529, 4.87484917945516723546545158294, 5.01564976002989168468776414543, 5.69208011630684437322158261218, 6.48451110526696282246961711325, 6.67009631001529545506185276841, 7.69164818258356184893378968692, 7.73831918709431562684545235915, 8.261177977090796325600765542566, 8.843078054411908501657519136744
