| L(s) = 1 | − 11·4-s + 28·7-s − 22·13-s + 45·16-s − 132·19-s − 308·28-s − 126·31-s − 354·37-s − 272·43-s + 490·49-s + 242·52-s − 1.17e3·61-s − 363·64-s − 38·67-s − 188·73-s + 1.45e3·76-s − 690·79-s − 616·91-s + 1.98e3·97-s − 1.98e3·103-s − 2.55e3·109-s + 1.26e3·112-s − 4.75e3·121-s + 1.38e3·124-s + 127-s + 131-s − 3.69e3·133-s + ⋯ |
| L(s) = 1 | − 1.37·4-s + 1.51·7-s − 0.469·13-s + 0.703·16-s − 1.59·19-s − 2.07·28-s − 0.730·31-s − 1.57·37-s − 0.964·43-s + 10/7·49-s + 0.645·52-s − 2.45·61-s − 0.708·64-s − 0.0692·67-s − 0.301·73-s + 2.19·76-s − 0.982·79-s − 0.709·91-s + 2.07·97-s − 1.89·103-s − 2.24·109-s + 1.06·112-s − 3.57·121-s + 1.00·124-s + 0.000698·127-s + 0.000666·131-s − 2.40·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 11 T^{2} + 19 p^{2} T^{4} + 11 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4757 T^{2} + 9131212 T^{4} + 4757 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 11 T + 334 p T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16103 T^{2} + 110752648 T^{4} + 16103 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 66 T + 762 p T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 16580 T^{2} + 227073517 T^{4} + 16580 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 57852 T^{2} + 1698017789 T^{4} + 57852 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 63 T + 42068 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 177 T + 90632 T^{2} + 177 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 93635 T^{2} + 8952796516 T^{4} + 93635 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 136 T + 147517 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 174744 T^{2} + 15146248718 T^{4} + 174744 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 494267 T^{2} + 102952583068 T^{4} + 494267 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 646299 T^{2} + 187556725712 T^{4} + 646299 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 585 T + 372962 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 19 T + 404134 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 397485 T^{2} + 173980338356 T^{4} + 397485 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 94 T + 606202 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 345 T + 1001934 T^{2} + 345 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 1768119 T^{2} + 1429880388068 T^{4} + 1768119 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 1398400 T^{2} + 1454863323486 T^{4} + 1398400 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 990 T + 1951602 T^{2} - 990 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.76529487309728492491636262408, −6.59271941974846036949428184158, −6.21133145015693867588509372853, −6.05631413059383073384896999307, −6.00143164801777872697499944006, −5.55685221795902748955977923649, −5.23102834602827771244710121409, −5.22088089529499081455848141947, −5.17004850143888157817590030889, −4.56563497805090906697797444809, −4.51038209326343143389820487526, −4.49170702716514902735483591877, −4.43962317105888616865172870694, −3.73589721903544306901852113696, −3.64966226670344065188603638656, −3.64839936346230240244506448210, −3.27150044087074957163924660323, −2.68392326501282368897616330100, −2.65622382088595926026509903834, −2.31724247039840670114315912482, −1.95332170470552353019557658833, −1.84447864536553326190305873046, −1.25115933733093710893864531613, −1.19764643516750001531035071499, −1.13093303974104489748271046879, 0, 0, 0, 0,
1.13093303974104489748271046879, 1.19764643516750001531035071499, 1.25115933733093710893864531613, 1.84447864536553326190305873046, 1.95332170470552353019557658833, 2.31724247039840670114315912482, 2.65622382088595926026509903834, 2.68392326501282368897616330100, 3.27150044087074957163924660323, 3.64839936346230240244506448210, 3.64966226670344065188603638656, 3.73589721903544306901852113696, 4.43962317105888616865172870694, 4.49170702716514902735483591877, 4.51038209326343143389820487526, 4.56563497805090906697797444809, 5.17004850143888157817590030889, 5.22088089529499081455848141947, 5.23102834602827771244710121409, 5.55685221795902748955977923649, 6.00143164801777872697499944006, 6.05631413059383073384896999307, 6.21133145015693867588509372853, 6.59271941974846036949428184158, 6.76529487309728492491636262408
