| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 2·9-s + 5·16-s − 4·18-s − 2·25-s + 6·32-s − 6·36-s − 4·50-s + 4·53-s + 7·64-s − 8·72-s + 3·81-s − 6·100-s + 8·106-s − 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s − 10·144-s + 149-s + 151-s + 157-s + 6·162-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.904559197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.904559197\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$ | \( ( 1 - T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−8.910798471797272718739241120261, −8.430633300389028200049488011712, −8.048066470631121885324039611231, −7.932166059349246756991103152183, −7.26929135940430910527311972030, −7.04115919702666259433507616079, −6.63573805565276938472088701074, −5.99821582395933511915893744867, −5.95268570667967206761893156753, −5.47196184114791683729290094088, −5.42316652431438731144289533028, −4.84989405626378519022573792706, −4.19642928662406139554867024405, −4.05546430443564320995967653251, −3.50207615625483771515222682221, −3.15546472061134942580040113142, −2.69363529614261304227035350493, −2.10603401765731893028781930880, −2.06773450539787846026591481102, −0.932916404743028582333596457171,
0.932916404743028582333596457171, 2.06773450539787846026591481102, 2.10603401765731893028781930880, 2.69363529614261304227035350493, 3.15546472061134942580040113142, 3.50207615625483771515222682221, 4.05546430443564320995967653251, 4.19642928662406139554867024405, 4.84989405626378519022573792706, 5.42316652431438731144289533028, 5.47196184114791683729290094088, 5.95268570667967206761893156753, 5.99821582395933511915893744867, 6.63573805565276938472088701074, 7.04115919702666259433507616079, 7.26929135940430910527311972030, 7.932166059349246756991103152183, 8.048066470631121885324039611231, 8.430633300389028200049488011712, 8.910798471797272718739241120261
