| L(s) = 1 | + 9-s + 2·13-s − 12·17-s − 10·25-s − 4·29-s + 20·37-s + 16·41-s − 10·49-s − 12·53-s + 4·61-s + 28·73-s + 81-s + 8·89-s + 28·97-s − 36·101-s − 4·109-s − 20·113-s + 2·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 0.554·13-s − 2.91·17-s − 2·25-s − 0.742·29-s + 3.28·37-s + 2.49·41-s − 1.42·49-s − 1.64·53-s + 0.512·61-s + 3.27·73-s + 1/9·81-s + 0.847·89-s + 2.84·97-s − 3.58·101-s − 0.383·109-s − 1.88·113-s + 0.184·117-s − 0.545·121-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 − 0.970·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1557504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.920126496569155684931452044745, −7.26948453317897544910090398161, −6.68084707610599017210864105435, −6.26794152529161447517798404710, −6.18680019324680760976289140185, −5.59232004512023197121827602220, −4.87228388162956512694940641649, −4.48413639840570454203249914552, −4.02331688083948425946247925030, −3.84092207155881713644722109378, −2.89365370501913595809144605675, −2.22728463189050631057902889299, −2.07382717156596942564746274979, −1.02393888089349693967293608434, 0,
1.02393888089349693967293608434, 2.07382717156596942564746274979, 2.22728463189050631057902889299, 2.89365370501913595809144605675, 3.84092207155881713644722109378, 4.02331688083948425946247925030, 4.48413639840570454203249914552, 4.87228388162956512694940641649, 5.59232004512023197121827602220, 6.18680019324680760976289140185, 6.26794152529161447517798404710, 6.68084707610599017210864105435, 7.26948453317897544910090398161, 7.920126496569155684931452044745
