L(s) = 1 | − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯ |
L(s) = 1 | − 2-s − 3·3-s + 5-s + 3·6-s + 4·7-s + 6·9-s − 10-s + 2·13-s − 4·14-s − 3·15-s − 6·18-s + 2·19-s − 12·21-s − 2·23-s − 2·26-s − 10·27-s + 3·30-s + 32-s + 4·35-s − 2·38-s − 6·39-s + 12·42-s + 6·45-s + 2·46-s + 10·49-s + 10·54-s − 6·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5360146764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5360146764\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 3 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 23 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 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$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 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_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 67 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12171918416976001221394753408, −6.63469588782169186030921665953, −6.43520394811542215432462433999, −6.25137562299408477533209363567, −6.18801518191244997682381618804, −5.87374430171563989399850384813, −5.69763042689353483256138357909, −5.54470601327486007760926593254, −5.42355156779689226948832990600, −5.06861614393463777531848420907, −4.90216729498878839613961551833, −4.63179191282320981710034938650, −4.38394864580764029181426475342, −4.37348790438039167033285866280, −4.35538968792354698104366005420, −3.53212327050473597218943166149, −3.49296635226964067475845602398, −3.34593507378023861084359138157, −2.30713421645267976560776689669, −2.14507414811926742479397129886, −2.00286324226684587564999816918, −1.43905388031223434604736478736, −1.36166945667014705849167415532, −1.25102279053126631379284794267, −0.856756275800495936277887822617,
0.856756275800495936277887822617, 1.25102279053126631379284794267, 1.36166945667014705849167415532, 1.43905388031223434604736478736, 2.00286324226684587564999816918, 2.14507414811926742479397129886, 2.30713421645267976560776689669, 3.34593507378023861084359138157, 3.49296635226964067475845602398, 3.53212327050473597218943166149, 4.35538968792354698104366005420, 4.37348790438039167033285866280, 4.38394864580764029181426475342, 4.63179191282320981710034938650, 4.90216729498878839613961551833, 5.06861614393463777531848420907, 5.42355156779689226948832990600, 5.54470601327486007760926593254, 5.69763042689353483256138357909, 5.87374430171563989399850384813, 6.18801518191244997682381618804, 6.25137562299408477533209363567, 6.43520394811542215432462433999, 6.63469588782169186030921665953, 7.12171918416976001221394753408