| L(s) = 1 | − 2·2-s + 2·4-s − 12·5-s − 4·7-s − 4·8-s + 18·9-s + 24·10-s − 12·11-s + 8·14-s + 8·16-s − 36·18-s − 52·19-s − 24·20-s + 24·22-s + 72·25-s − 8·28-s + 96·29-s − 56·31-s − 8·32-s + 48·35-s + 36·36-s − 74·37-s + 104·38-s + 48·40-s + 18·41-s − 24·44-s − 216·45-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 2.39·5-s − 4/7·7-s − 1/2·8-s + 2·9-s + 12/5·10-s − 1.09·11-s + 4/7·14-s + 1/2·16-s − 2·18-s − 2.73·19-s − 6/5·20-s + 1.09·22-s + 2.87·25-s − 2/7·28-s + 3.31·29-s − 1.80·31-s − 1/4·32-s + 1.37·35-s + 36-s − 2·37-s + 2.73·38-s + 6/5·40-s + 0.439·41-s − 0.545·44-s − 4.79·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3032364205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3032364205\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 4 T + 8 T^{2} - 360 T^{3} - 3121 T^{4} - 360 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} - 2040 T^{3} - 26881 T^{4} - 2040 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 542 T^{2} + 210243 T^{4} + 542 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 52 T + 1352 T^{2} + 32760 T^{3} + 721439 T^{4} + 32760 p^{2} T^{5} + 1352 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 482 T^{2} - 47517 T^{4} + 482 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 48 T + 1463 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 28 T + 392 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T + p^{2} T^{2} )^{2}( 1 - p^{2} T^{2} + p^{4} T^{4} ) \) |
| 41 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} + 57600 T^{3} - 3344161 T^{4} + 57600 p^{2} T^{5} + 162 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^3$ | \( 1 + 2402 T^{2} + 2350803 T^{4} + 2402 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 108 T + 5832 T^{2} + 122040 T^{3} - 18707521 T^{4} + 122040 p^{2} T^{5} + 5832 p^{4} T^{6} - 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 18 T - 3397 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - 44 T + 968 T^{2} + 352440 T^{3} - 27904801 T^{4} + 352440 p^{2} T^{5} + 968 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 12 T + 72 T^{2} - 120120 T^{3} - 26132401 T^{4} - 120120 p^{2} T^{5} + 72 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 108 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 156 T + 12168 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 18 T + 162 T^{2} + 282240 T^{3} - 65282401 T^{4} + 282240 p^{2} T^{5} + 162 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $C_2^3$ | \( 1 - 94 T + 4418 T^{2} + 1353600 T^{3} - 152148481 T^{4} + 1353600 p^{2} T^{5} + 4418 p^{4} T^{6} - 94 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.325336974031626909192131502635, −7.88394916350271449385967790260, −7.60268480085748439196547282787, −7.47774250793877394269889501774, −7.17787195961130728154076810087, −6.88934902499798843098060106992, −6.80375992941791513601081968349, −6.78776706339074716900467221297, −6.21263016425121511723422407487, −5.77432942353784332594722587847, −5.63761306732714835126888793305, −5.22093677650847199515781006721, −4.79589425395880233235867020766, −4.48408917798117936062220316781, −4.29423719247434637804522552562, −4.02332793436250553759373833282, −3.83896192752058996785142553693, −3.57265013422337033734331350794, −3.10056428831029367857258307506, −2.52259459541151749065638992408, −2.37755498528542697382829607215, −1.97880252384278742705133107964, −1.09926410071360704099714243896, −0.78027029790163845958258569386, −0.22156420697015050367786911841,
0.22156420697015050367786911841, 0.78027029790163845958258569386, 1.09926410071360704099714243896, 1.97880252384278742705133107964, 2.37755498528542697382829607215, 2.52259459541151749065638992408, 3.10056428831029367857258307506, 3.57265013422337033734331350794, 3.83896192752058996785142553693, 4.02332793436250553759373833282, 4.29423719247434637804522552562, 4.48408917798117936062220316781, 4.79589425395880233235867020766, 5.22093677650847199515781006721, 5.63761306732714835126888793305, 5.77432942353784332594722587847, 6.21263016425121511723422407487, 6.78776706339074716900467221297, 6.80375992941791513601081968349, 6.88934902499798843098060106992, 7.17787195961130728154076810087, 7.47774250793877394269889501774, 7.60268480085748439196547282787, 7.88394916350271449385967790260, 8.325336974031626909192131502635
