| L(s) = 1 | + 2·3-s + 9-s + 2·17-s + 3·19-s − 25-s + 2·41-s − 2·43-s − 49-s + 4·51-s + 6·57-s − 3·59-s + 2·67-s + 2·73-s − 2·75-s + 3·83-s − 2·89-s + 3·97-s − 2·107-s + 3·113-s + 4·123-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2·3-s + 9-s + 2·17-s + 3·19-s − 25-s + 2·41-s − 2·43-s − 49-s + 4·51-s + 6·57-s − 3·59-s + 2·67-s + 2·73-s − 2·75-s + 3·83-s − 2·89-s + 3·97-s − 2·107-s + 3·113-s + 4·123-s + 127-s − 4·129-s + 131-s + 137-s + 139-s − 2·147-s + 149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.932532340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.932532340\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.35745435065815379738462638080, −5.72689167878600410934667941770, −5.68153240748925409211803079870, −5.63422276569278538353517909849, −5.58572892118339924339937804363, −5.04273341074800974840199231103, −5.02139650787599042573495607573, −4.83793405676695470512582411823, −4.66586230065132732679243392519, −4.34594910373285429935683381437, −3.96870985714845225833367544956, −3.85182737127358611040557557023, −3.46318401422789190775104227816, −3.45854432948231624950666662606, −3.22590854936514227551528630136, −3.16709448457550771719738425193, −3.01917560935157414383176941724, −2.66603178118192682379920509198, −2.38101477826994655600343650327, −2.26136258499284032364742265755, −1.85827543193260989379922506009, −1.64341136393333286695400109638, −1.30769765556086489800866136470, −0.942487295110757282815066675372, −0.77495621543735356622371738147,
0.77495621543735356622371738147, 0.942487295110757282815066675372, 1.30769765556086489800866136470, 1.64341136393333286695400109638, 1.85827543193260989379922506009, 2.26136258499284032364742265755, 2.38101477826994655600343650327, 2.66603178118192682379920509198, 3.01917560935157414383176941724, 3.16709448457550771719738425193, 3.22590854936514227551528630136, 3.45854432948231624950666662606, 3.46318401422789190775104227816, 3.85182737127358611040557557023, 3.96870985714845225833367544956, 4.34594910373285429935683381437, 4.66586230065132732679243392519, 4.83793405676695470512582411823, 5.02139650787599042573495607573, 5.04273341074800974840199231103, 5.58572892118339924339937804363, 5.63422276569278538353517909849, 5.68153240748925409211803079870, 5.72689167878600410934667941770, 6.35745435065815379738462638080
