| L(s) = 1 | + 4·3-s − 4·4-s + 8·9-s − 16·12-s − 96·13-s + 8·16-s + 16·19-s − 12·27-s + 72·31-s − 32·36-s + 112·37-s − 384·39-s − 240·43-s + 32·48-s + 360·49-s + 384·52-s + 64·57-s + 208·61-s − 64·64-s − 232·67-s − 64·76-s − 136·79-s − 34·81-s + 288·93-s − 480·97-s + 48·108-s + 64·109-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 4-s + 8/9·9-s − 4/3·12-s − 7.38·13-s + 1/2·16-s + 0.842·19-s − 4/9·27-s + 2.32·31-s − 8/9·36-s + 3.02·37-s − 9.84·39-s − 5.58·43-s + 2/3·48-s + 7.34·49-s + 7.38·52-s + 1.12·57-s + 3.40·61-s − 64-s − 3.46·67-s − 0.842·76-s − 1.72·79-s − 0.419·81-s + 3.09·93-s − 4.94·97-s + 4/9·108-s + 0.587·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8225113089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8225113089\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T^{2} + p^{3} T^{4} + p^{6} T^{6} + p^{8} T^{8} \) |
| 3 | \( 1 - 4 T + 8 T^{2} + 4 p T^{3} - 14 p^{2} T^{4} + 4 p^{3} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 5 | \( 1 + 372 T^{4} + 464614 T^{8} + 372 p^{8} T^{12} + p^{16} T^{16} \) |
| 7 | \( ( 1 - 180 T^{2} + 12874 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 + 37380 T^{4} + 692870854 T^{8} + 37380 p^{8} T^{12} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 48 T + 1152 T^{2} + 21264 T^{3} + 317422 T^{4} + 21264 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 236 T^{2} + 73334 T^{4} - 236 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 8 T + 32 T^{2} - 1160 T^{3} - 4606 T^{4} - 1160 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( ( 1 + 996 T^{2} + 719878 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 147060 T^{4} + 623812106854 T^{8} + 147060 p^{8} T^{12} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 18 T + 1996 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 56 T + 1568 T^{2} - 79016 T^{3} + 3980078 T^{4} - 79016 p^{2} T^{5} + 1568 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( ( 1 + 6060 T^{2} + 14724790 T^{4} + 6060 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 + 120 T + 7200 T^{2} + 411000 T^{3} + 20977474 T^{4} + 411000 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 692 T^{2} + 7491686 T^{4} - 692 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( 1 + 1441524 T^{4} - 91064986479386 T^{8} + 1441524 p^{8} T^{12} + p^{16} T^{16} \) |
| 59 | \( 1 + 15787044 T^{4} + 346992256453126 T^{8} + 15787044 p^{8} T^{12} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 104 T + 5408 T^{2} - 491192 T^{3} + 43609454 T^{4} - 491192 p^{2} T^{5} + 5408 p^{4} T^{6} - 104 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 + 116 T + 6728 T^{2} + 350436 T^{3} + 16097858 T^{4} + 350436 p^{2} T^{5} + 6728 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 + 3604 T^{2} + 10180454 T^{4} + 3604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 17604 T^{2} + 131615494 T^{4} - 17604 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 + 34 T + 11588 T^{2} + 34 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( 1 - 64624380 T^{4} + 2364945173919814 T^{8} - 64624380 p^{8} T^{12} + p^{16} T^{16} \) |
| 89 | \( ( 1 + 25444 T^{2} + 277784198 T^{4} + 25444 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 120 T + 21970 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
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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
−7.25868542493107346723300485411, −7.17379882616971866445988525844, −7.08046938714429968738109920332, −7.04628767577453036506155257882, −6.90300451541053274614993403227, −6.71320460557123378436311171871, −6.17518696605526415138512955089, −5.79456218876992632961998236109, −5.71654063438368055418911317529, −5.45655395650923373261364631348, −5.31427078937959112917603791232, −5.00628057703952830532706431236, −4.95915352064354997919134315054, −4.58093625976878605689822281084, −4.55693522189325662923867006818, −4.22827261061241541052792511477, −4.07396037133073444669557435040, −3.92938028915399901188281093598, −2.98152972252694508629880344077, −2.89884597339543381502222992146, −2.88242278752872149379223778612, −2.55914758882068127390786325796, −2.27144330983749714748486608558, −1.92480159761492582684069068044, −0.48565899256073123151709289078,
0.48565899256073123151709289078, 1.92480159761492582684069068044, 2.27144330983749714748486608558, 2.55914758882068127390786325796, 2.88242278752872149379223778612, 2.89884597339543381502222992146, 2.98152972252694508629880344077, 3.92938028915399901188281093598, 4.07396037133073444669557435040, 4.22827261061241541052792511477, 4.55693522189325662923867006818, 4.58093625976878605689822281084, 4.95915352064354997919134315054, 5.00628057703952830532706431236, 5.31427078937959112917603791232, 5.45655395650923373261364631348, 5.71654063438368055418911317529, 5.79456218876992632961998236109, 6.17518696605526415138512955089, 6.71320460557123378436311171871, 6.90300451541053274614993403227, 7.04628767577453036506155257882, 7.08046938714429968738109920332, 7.17379882616971866445988525844, 7.25868542493107346723300485411
Plot not available for L-functions of degree greater than 10.
