| L(s) = 1 | − 8·13-s + 32·25-s + 48·37-s + 40·49-s − 32·61-s − 96·73-s + 32·97-s + 24·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 2.21·13-s + 32/5·25-s + 7.89·37-s + 40/7·49-s − 4.09·61-s − 11.2·73-s + 3.24·97-s + 2.29·109-s − 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.26959396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.26959396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 - 16 T^{2} + 111 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 20 T^{2} + 186 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 16 T^{2} + 114 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 - 56 T^{2} + 1335 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 40 T^{2} + 2007 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 68 T^{2} + 2310 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 136 T^{2} + 7794 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 76 T^{2} + 4662 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 100 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 68 T^{2} + 6918 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 8 T + 111 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 28 T^{2} + 8586 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 152 T^{2} + 14130 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 24 T + 287 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 172 T^{2} + 18906 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 220 T^{2} + 22806 T^{4} + 220 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 128 T^{2} + 8031 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.16398766667903062052870269044, −3.03969115847942410893393363585, −3.03037343701486716326408856786, −2.93967134361160102205877263644, −2.83370462277742226236326965543, −2.82687727854987411813108799857, −2.69620839392640509349813127375, −2.66235279858839135698906567389, −2.53277128872875887941440238385, −2.43542479076989545663079567764, −2.31481062896495885493343332638, −2.21952613169529739552727386253, −2.12867067442407274136352927433, −1.78661209937616048470810355157, −1.50792570367219109194890040996, −1.48096135660787156595538028525, −1.46149749950985425748995324597, −1.16146326582422151452989718248, −1.13237270730880841615143767565, −1.09782282782195024766945104906, −0.914287708961009985560697375064, −0.55986025436932480098129400539, −0.55507175510655696424041951821, −0.50453522316932575411769310788, −0.17443457264764364745125563895,
0.17443457264764364745125563895, 0.50453522316932575411769310788, 0.55507175510655696424041951821, 0.55986025436932480098129400539, 0.914287708961009985560697375064, 1.09782282782195024766945104906, 1.13237270730880841615143767565, 1.16146326582422151452989718248, 1.46149749950985425748995324597, 1.48096135660787156595538028525, 1.50792570367219109194890040996, 1.78661209937616048470810355157, 2.12867067442407274136352927433, 2.21952613169529739552727386253, 2.31481062896495885493343332638, 2.43542479076989545663079567764, 2.53277128872875887941440238385, 2.66235279858839135698906567389, 2.69620839392640509349813127375, 2.82687727854987411813108799857, 2.83370462277742226236326965543, 2.93967134361160102205877263644, 3.03037343701486716326408856786, 3.03969115847942410893393363585, 3.16398766667903062052870269044
Plot not available for L-functions of degree greater than 10.
