| L(s) = 1 | − 3-s − 2·4-s − 3·7-s + 2·12-s + 3·13-s + 3·16-s + 19-s + 3·21-s + 25-s + 27-s + 6·28-s − 2·31-s + 2·37-s − 3·39-s + 2·43-s − 3·48-s + 5·49-s − 6·52-s − 57-s − 3·61-s − 4·64-s + 2·67-s + 73-s − 75-s − 2·76-s − 79-s − 81-s + ⋯ |
| L(s) = 1 | − 3-s − 2·4-s − 3·7-s + 2·12-s + 3·13-s + 3·16-s + 19-s + 3·21-s + 25-s + 27-s + 6·28-s − 2·31-s + 2·37-s − 3·39-s + 2·43-s − 3·48-s + 5·49-s − 6·52-s − 57-s − 3·61-s − 4·64-s + 2·67-s + 73-s − 75-s − 2·76-s − 79-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1418481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1418481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3075944817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3075944817\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 397 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.910804776469886212801322055978, −9.719650491537961470731312083948, −9.281008776609150354845338294198, −9.112086111019762132601855847828, −8.698895641271596281357618521882, −8.423800458865870935332890082360, −7.61901046915063473830617132673, −7.32506943170227492889266914057, −6.41477002112533212718053879528, −6.35862807686730366278487955415, −5.86500816419121904623155290348, −5.74841708359112710365280960320, −5.23803889531485341861914378000, −4.48797240556585259461349637436, −3.93147918738295870543552061143, −3.70031432019657060922039545343, −3.18975106289151085018506111104, −2.93245060061195969706874171785, −1.15323466779719063038657096547, −0.66669576585410825697822313884,
0.66669576585410825697822313884, 1.15323466779719063038657096547, 2.93245060061195969706874171785, 3.18975106289151085018506111104, 3.70031432019657060922039545343, 3.93147918738295870543552061143, 4.48797240556585259461349637436, 5.23803889531485341861914378000, 5.74841708359112710365280960320, 5.86500816419121904623155290348, 6.35862807686730366278487955415, 6.41477002112533212718053879528, 7.32506943170227492889266914057, 7.61901046915063473830617132673, 8.423800458865870935332890082360, 8.698895641271596281357618521882, 9.112086111019762132601855847828, 9.281008776609150354845338294198, 9.719650491537961470731312083948, 9.910804776469886212801322055978
