| L(s) = 1 | + 8·13-s − 16·17-s − 16·29-s + 24·37-s + 8·49-s + 48·53-s + 24·61-s − 32·97-s − 16·101-s + 24·109-s − 16·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2.21·13-s − 3.88·17-s − 2.97·29-s + 3.94·37-s + 8/7·49-s + 6.59·53-s + 3.07·61-s − 3.24·97-s − 1.59·101-s + 2.29·109-s − 1.50·113-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 + 2.46·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.034139460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.034139460\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 - p T^{2} )^{4}( 1 + p T^{2} )^{4} \) |
| 7 | \( ( 1 - 4 T^{2} + 22 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 + 164 T^{4} + 30886 T^{8} + 164 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 - 412 T^{4} + 52198 T^{8} - 412 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 8 T + 32 T^{2} + 216 T^{3} + 1454 T^{4} + 216 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T^{2} - 74 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 72 T^{2} - 180 T^{3} - 34 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 116 T^{2} + 6406 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 164 T^{4} - 3523674 T^{8} + 164 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( ( 1 + p T^{2} )^{8} \) |
| 53 | \( ( 1 - 24 T + 288 T^{2} - 2760 T^{3} + 22606 T^{4} - 2760 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 5518 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 72 T^{2} - 468 T^{3} + 2558 T^{4} - 468 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8878 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 188 T^{2} + 17638 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 60 T^{2} + 38 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 260 T^{2} + 28662 T^{4} + 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 13412 T^{4} + 114077158 T^{8} + 13412 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 16 T + p T^{2} )^{4}( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.57805815096761593860060794645, −3.46803603867532306035375158688, −3.00686158182531924479703857288, −2.94983043080666218199127587640, −2.90449331206207167048912705435, −2.89454623819484194010072713512, −2.61830281478256907262126090708, −2.58934859316200617990556008698, −2.43591304560280995268232450140, −2.43102926856535422641256012636, −2.31411655084975724817799231177, −2.10118089584115989723021360105, −2.04420061358294802197347790833, −1.82790746508723381767135996511, −1.78242631399437181611686714641, −1.69055145384047169342843646587, −1.63087641908415608865315017363, −1.27573351259252686247057398408, −1.06209092593778695801302096657, −0.917105858888031127374870371013, −0.811103565947265335942213973914, −0.77313771812318157624999144496, −0.66231907442727197480794022647, −0.32233182859220640735170375509, −0.13555447661746361159902003890,
0.13555447661746361159902003890, 0.32233182859220640735170375509, 0.66231907442727197480794022647, 0.77313771812318157624999144496, 0.811103565947265335942213973914, 0.917105858888031127374870371013, 1.06209092593778695801302096657, 1.27573351259252686247057398408, 1.63087641908415608865315017363, 1.69055145384047169342843646587, 1.78242631399437181611686714641, 1.82790746508723381767135996511, 2.04420061358294802197347790833, 2.10118089584115989723021360105, 2.31411655084975724817799231177, 2.43102926856535422641256012636, 2.43591304560280995268232450140, 2.58934859316200617990556008698, 2.61830281478256907262126090708, 2.89454623819484194010072713512, 2.90449331206207167048912705435, 2.94983043080666218199127587640, 3.00686158182531924479703857288, 3.46803603867532306035375158688, 3.57805815096761593860060794645
Plot not available for L-functions of degree greater than 10.
