| L(s) = 1 | − 12·3-s + 90·9-s + 40·11-s − 120·17-s − 320·25-s − 540·27-s − 480·33-s − 552·41-s + 432·43-s − 832·49-s + 1.44e3·51-s + 1.36e3·59-s + 864·67-s − 240·73-s + 3.84e3·75-s + 2.83e3·81-s + 952·83-s − 1.56e3·89-s − 2.28e3·97-s + 3.60e3·99-s + 944·107-s − 4.20e3·113-s − 4.18e3·121-s + 6.62e3·123-s + 127-s − 5.18e3·129-s + 131-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 10/3·9-s + 1.09·11-s − 1.71·17-s − 2.55·25-s − 3.84·27-s − 2.53·33-s − 2.10·41-s + 1.53·43-s − 2.42·49-s + 3.95·51-s + 3.00·59-s + 1.57·67-s − 0.384·73-s + 5.91·75-s + 35/9·81-s + 1.25·83-s − 1.85·89-s − 2.38·97-s + 3.65·99-s + 0.852·107-s − 3.49·113-s − 3.14·121-s + 4.85·123-s + 0.000698·127-s − 3.53·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 64 p T^{2} + 1986 p^{2} T^{4} + 64 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 832 T^{2} + 7746 p^{2} T^{4} + 832 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 20 T + 2690 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8428 T^{2} + 27382614 T^{4} + 8428 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 60 T + 10334 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 12750 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 7628 T^{2} - 37861626 T^{4} + 7628 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 21056 T^{2} + 1079985426 T^{4} + 21056 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 75424 T^{2} + 3008245506 T^{4} + 75424 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 106492 T^{2} + 7210762134 T^{4} + 106492 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 276 T + 155086 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 216 T + 146478 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132332 T^{2} + 9563109414 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 560768 T^{2} + 122848689714 T^{4} + 560768 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 680 T + 526070 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 345244 T^{2} + 119910783606 T^{4} + 345244 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 432 T + 416982 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1199084 T^{2} + 615208348806 T^{4} + 1199084 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 120 T + 707906 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1369696 T^{2} + 864453694146 T^{4} + 1369696 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 476 T + 1155218 T^{2} - 476 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 780 T + 1357238 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1140 T + 2147654 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.81296491459586454501287937567, −6.51727298103880857729355336959, −6.25826601748738066284960159665, −6.25764140663068494224031894586, −6.06787131882568743726502840872, −5.48905223919425017090801160753, −5.44613655595055717246737212985, −5.36650515900072003208486498036, −5.34725689197971040587700203144, −4.76420579472315199396130998699, −4.56356458945280725716976829604, −4.49515007146384190261256276092, −4.26752578888643032578975017192, −3.81681831515155398131454395709, −3.72958576316506427797392667382, −3.58349347193319048362197349778, −3.53213749023341301606250630228, −2.60325687591277239813422242193, −2.56706265034843121058592092930, −2.21112396622871517999062073378, −2.13686576725244597829911665922, −1.52161156874284673136164618537, −1.38883984209201664146423456038, −1.07823196081252394166041858523, −1.00661677991681724631040277494, 0, 0, 0, 0,
1.00661677991681724631040277494, 1.07823196081252394166041858523, 1.38883984209201664146423456038, 1.52161156874284673136164618537, 2.13686576725244597829911665922, 2.21112396622871517999062073378, 2.56706265034843121058592092930, 2.60325687591277239813422242193, 3.53213749023341301606250630228, 3.58349347193319048362197349778, 3.72958576316506427797392667382, 3.81681831515155398131454395709, 4.26752578888643032578975017192, 4.49515007146384190261256276092, 4.56356458945280725716976829604, 4.76420579472315199396130998699, 5.34725689197971040587700203144, 5.36650515900072003208486498036, 5.44613655595055717246737212985, 5.48905223919425017090801160753, 6.06787131882568743726502840872, 6.25764140663068494224031894586, 6.25826601748738066284960159665, 6.51727298103880857729355336959, 6.81296491459586454501287937567
