L(s) = 1 | + 3-s − 4·4-s − 5-s + 7-s − 2·9-s − 4·12-s + 10·13-s − 15-s + 12·16-s + 4·20-s + 21-s − 12·23-s + 25-s − 5·27-s − 4·28-s − 35-s + 8·36-s + 10·39-s − 24·41-s + 2·45-s + 12·48-s + 49-s − 40·52-s + 24·53-s + 4·60-s − 2·63-s − 32·64-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2·4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.15·12-s + 2.77·13-s − 0.258·15-s + 3·16-s + 0.894·20-s + 0.218·21-s − 2.50·23-s + 1/5·25-s − 0.962·27-s − 0.755·28-s − 0.169·35-s + 4/3·36-s + 1.60·39-s − 3.74·41-s + 0.298·45-s + 1.73·48-s + 1/7·49-s − 5.54·52-s + 3.29·53-s + 0.516·60-s − 0.251·63-s − 4·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385875 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−8.560437390952492594261182815487, −8.190558210269945144592569148827, −7.980947296320422939728903060714, −7.22517929037828263493597858765, −6.40352559661425609695603895056, −5.98285749598845169856705301545, −5.47907564040983295563221596641, −5.18210179787882528820992169724, −4.25071896913475536050876516289, −3.97193527118658395993579674398, −3.61689003045514298236484607491, −3.21404721245224231470064978199, −1.96292462969238605585350779017, −1.14803979455536762352526385352, 0,
1.14803979455536762352526385352, 1.96292462969238605585350779017, 3.21404721245224231470064978199, 3.61689003045514298236484607491, 3.97193527118658395993579674398, 4.25071896913475536050876516289, 5.18210179787882528820992169724, 5.47907564040983295563221596641, 5.98285749598845169856705301545, 6.40352559661425609695603895056, 7.22517929037828263493597858765, 7.980947296320422939728903060714, 8.190558210269945144592569148827, 8.560437390952492594261182815487