| L(s) = 1 | − 3·4-s + 12·7-s − 32·13-s + 16·16-s − 8·19-s + 5·25-s − 36·28-s + 36·31-s − 64·37-s − 32·43-s + 134·49-s + 96·52-s − 164·61-s − 117·64-s − 48·67-s − 296·73-s + 24·76-s − 276·79-s − 384·91-s + 332·97-s − 15·100-s − 52·103-s + 152·109-s + 192·112-s − 222·121-s − 108·124-s + 127-s + ⋯ |
| L(s) = 1 | − 3/4·4-s + 12/7·7-s − 2.46·13-s + 16-s − 0.421·19-s + 1/5·25-s − 9/7·28-s + 1.16·31-s − 1.72·37-s − 0.744·43-s + 2.73·49-s + 1.84·52-s − 2.68·61-s − 1.82·64-s − 0.716·67-s − 4.05·73-s + 6/19·76-s − 3.49·79-s − 4.21·91-s + 3.42·97-s − 0.149·100-s − 0.504·103-s + 1.39·109-s + 12/7·112-s − 1.83·121-s − 0.870·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1818875151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1818875151\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} - 7 T^{4} + 3 p^{4} T^{6} + p^{8} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T - 13 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 222 T^{2} + 34643 T^{4} + 222 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 16 T + 87 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 558 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 44 T + 1407 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )( 1 + 44 T + 1407 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 29 | $C_2^3$ | \( 1 + 702 T^{2} - 214477 T^{4} + 702 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 18 T - 637 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 558 T^{2} - 2514397 T^{4} - 558 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 16 T - 1593 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1998 T^{2} - 887677 T^{4} + 1998 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 6942 T^{2} + 36074003 T^{4} + 6942 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 82 T + 3003 T^{2} + 82 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 24 T - 3913 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5598 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 138 T + 12803 T^{2} + 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 4958 T^{2} - 22876557 T^{4} + 4958 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2}( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 166 T + 18147 T^{2} - 166 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.88486131985701260510249919584, −7.87711990774809287071946415055, −7.33886291828441924473253920519, −7.14215922317609484793223257465, −7.10925933311462358363762477834, −7.04291992831932314593912657488, −6.20089988989085521849222319231, −6.06975599757454383033116494567, −5.94829811397003945995900373017, −5.57687945345266415416219105165, −5.10539499472028878433399908527, −5.09173139067592330260683461908, −4.84292658764540987784707405111, −4.44513144876836018399967967255, −4.30170947849574776750268643767, −4.30114065838001335073272422810, −3.66750442383478570843400769290, −3.01380016694650550325465066719, −2.94248267544347463772944861776, −2.83592150039026983523260899663, −2.01565948572333217423528775886, −1.80995418471933533059442319523, −1.51832174230599798408075286416, −0.924880755090013579439579902480, −0.094264536159551397494271509950,
0.094264536159551397494271509950, 0.924880755090013579439579902480, 1.51832174230599798408075286416, 1.80995418471933533059442319523, 2.01565948572333217423528775886, 2.83592150039026983523260899663, 2.94248267544347463772944861776, 3.01380016694650550325465066719, 3.66750442383478570843400769290, 4.30114065838001335073272422810, 4.30170947849574776750268643767, 4.44513144876836018399967967255, 4.84292658764540987784707405111, 5.09173139067592330260683461908, 5.10539499472028878433399908527, 5.57687945345266415416219105165, 5.94829811397003945995900373017, 6.06975599757454383033116494567, 6.20089988989085521849222319231, 7.04291992831932314593912657488, 7.10925933311462358363762477834, 7.14215922317609484793223257465, 7.33886291828441924473253920519, 7.87711990774809287071946415055, 7.88486131985701260510249919584
