L(s) = 1 | − 2·3-s + 3·9-s − 4·11-s + 4·17-s − 8·19-s − 8·25-s − 4·27-s + 8·33-s + 12·41-s − 16·43-s − 12·49-s − 8·51-s + 16·57-s − 24·59-s − 16·67-s − 16·73-s + 16·75-s + 5·81-s − 12·83-s + 4·89-s − 28·97-s − 12·99-s − 8·107-s + 12·113-s − 10·121-s − 24·123-s + 127-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 9-s − 1.20·11-s + 0.970·17-s − 1.83·19-s − 8/5·25-s − 0.769·27-s + 1.39·33-s + 1.87·41-s − 2.43·43-s − 1.71·49-s − 1.12·51-s + 2.11·57-s − 3.12·59-s − 1.95·67-s − 1.87·73-s + 1.84·75-s + 5/9·81-s − 1.31·83-s + 0.423·89-s − 2.84·97-s − 1.20·99-s − 0.773·107-s + 1.12·113-s − 0.909·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−9.348312489030541464786847511639, −8.916750607396651948811808976466, −8.199730259054469365479242027210, −8.137893719831541592670922312559, −7.47863373521185848634725783901, −7.47711933085206353286468171164, −6.67272090731286239914611731222, −6.25303693094547438036666320109, −6.01749574579307462209372075371, −5.67131832040511532506282615930, −4.98878792918103372545787665327, −4.88681667454852198314461845105, −4.09930114606367679589254130454, −4.02375112447157306290108008533, −2.95338886371393166452461882867, −2.82282625775304883596636989220, −1.69252627773279490791676290783, −1.56463936515313379078387460553, 0, 0,
1.56463936515313379078387460553, 1.69252627773279490791676290783, 2.82282625775304883596636989220, 2.95338886371393166452461882867, 4.02375112447157306290108008533, 4.09930114606367679589254130454, 4.88681667454852198314461845105, 4.98878792918103372545787665327, 5.67131832040511532506282615930, 6.01749574579307462209372075371, 6.25303693094547438036666320109, 6.67272090731286239914611731222, 7.47711933085206353286468171164, 7.47863373521185848634725783901, 8.137893719831541592670922312559, 8.199730259054469365479242027210, 8.916750607396651948811808976466, 9.348312489030541464786847511639