L(s) = 1 | + 4·5-s − 6·9-s − 4·11-s + 2·25-s + 16·31-s − 4·37-s − 24·45-s − 16·47-s + 49-s + 12·53-s − 16·55-s − 8·67-s − 16·71-s + 27·81-s − 12·89-s − 12·97-s + 24·99-s − 32·103-s + 4·113-s + 5·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 2·9-s − 1.20·11-s + 2/5·25-s + 2.87·31-s − 0.657·37-s − 3.57·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s − 2.15·55-s − 0.977·67-s − 1.89·71-s + 3·81-s − 1.27·89-s − 1.21·97-s + 2.41·99-s − 3.15·103-s + 0.376·113-s + 5/11·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−8.423245666401170068214022844193, −8.188420417684287628984093773695, −7.69587064441186150768078543731, −6.88310002274215763839569601337, −6.42262084813563967264199691637, −5.94974241082854154889853475745, −5.72315905059004231786170242312, −5.22316409435338364257345412913, −4.86884684581528043387154729760, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.79183800612725741835263061431, −2.30074942792092931708066021417, −1.44245425010921288807179765849, 0,
1.44245425010921288807179765849, 2.30074942792092931708066021417, 2.79183800612725741835263061431, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.86884684581528043387154729760, 5.22316409435338364257345412913, 5.72315905059004231786170242312, 5.94974241082854154889853475745, 6.42262084813563967264199691637, 6.88310002274215763839569601337, 7.69587064441186150768078543731, 8.188420417684287628984093773695, 8.423245666401170068214022844193