| L(s) = 1 | − 162·9-s + 644·17-s + 1.05e3·25-s + 6.07e3·41-s + 4.80e3·49-s − 2.88e3·73-s + 1.96e4·81-s + 1.95e4·89-s − 3.83e3·97-s − 4.92e4·113-s − 2.92e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.04e5·153-s + 157-s + 163-s + 167-s + 478·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·9-s + 2.22·17-s + 1.68·25-s + 3.61·41-s + 2·49-s − 0.541·73-s + 3·81-s + 2.46·89-s − 0.407·97-s − 3.85·113-s − 2·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 4.45·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.0167·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.554203703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554203703\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 48 T + p^{4} T^{2} )( 1 + 48 T + p^{4} T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 240 T + p^{4} T^{2} )( 1 + 240 T + p^{4} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 322 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1680 T + p^{4} T^{2} )( 1 + 1680 T + p^{4} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1680 T + p^{4} T^{2} )( 1 + 1680 T + p^{4} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3038 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 5040 T + p^{4} T^{2} )( 1 + 5040 T + p^{4} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2640 T + p^{4} T^{2} )( 1 + 2640 T + p^{4} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1442 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9758 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1918 T + p^{4} T^{2} )^{2} \) |
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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
−11.66997074714927005833523711048, −11.07032087001438903836652838168, −10.57431779433003655574526218514, −10.47921393782052633276704608620, −9.366110857361754624671600045510, −9.342551979106282954328568564597, −8.708460160415300390012574528809, −8.215051650848447844845988763648, −7.63596499892832766294666029863, −7.40168820764633063015907403034, −6.37834018800239921539430096967, −6.04697724797710366525334362999, −5.33823099670255678200582233246, −5.27455273628916884493677674277, −4.18630749664947921581523765470, −3.53326082059487626987666624549, −2.73374474756884502958949138725, −2.60229725484016872083284143998, −1.12685120835167801322494201599, −0.61197829506278032947422940665,
0.61197829506278032947422940665, 1.12685120835167801322494201599, 2.60229725484016872083284143998, 2.73374474756884502958949138725, 3.53326082059487626987666624549, 4.18630749664947921581523765470, 5.27455273628916884493677674277, 5.33823099670255678200582233246, 6.04697724797710366525334362999, 6.37834018800239921539430096967, 7.40168820764633063015907403034, 7.63596499892832766294666029863, 8.215051650848447844845988763648, 8.708460160415300390012574528809, 9.342551979106282954328568564597, 9.366110857361754624671600045510, 10.47921393782052633276704608620, 10.57431779433003655574526218514, 11.07032087001438903836652838168, 11.66997074714927005833523711048
