| L(s) = 1 | + 2·2-s − 4-s − 4·5-s − 10·7-s − 4·8-s − 8·10-s − 20·14-s + 2·17-s − 16·19-s + 4·20-s + 10·23-s + 10·25-s + 10·28-s − 8·29-s − 8·31-s + 2·32-s + 4·34-s + 40·35-s + 2·37-s − 32·38-s + 16·40-s + 8·41-s − 2·43-s + 20·46-s + 8·47-s + 42·49-s + 20·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 1.78·5-s − 3.77·7-s − 1.41·8-s − 2.52·10-s − 5.34·14-s + 0.485·17-s − 3.67·19-s + 0.894·20-s + 2.08·23-s + 2·25-s + 1.88·28-s − 1.48·29-s − 1.43·31-s + 0.353·32-s + 0.685·34-s + 6.76·35-s + 0.328·37-s − 5.19·38-s + 2.52·40-s + 1.24·41-s − 0.304·43-s + 2.94·46-s + 1.16·47-s + 6·49-s + 2.82·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - p^{4} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 58 T^{2} + 232 T^{3} + 703 T^{4} + 232 p T^{5} + 58 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 14 T^{2} + 9 p T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T + 50 T^{2} - 92 T^{3} + 1135 T^{4} - 92 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 8 T + 51 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 98 T^{2} - 544 T^{3} + 137 p T^{4} - 544 p T^{5} + 98 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 98 T^{2} + 656 T^{3} + 4003 T^{4} + 656 p T^{5} + 98 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 94 T^{2} - 260 T^{3} + 4219 T^{4} - 260 p T^{5} + 94 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 83 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 154 T^{2} + 248 T^{3} + 9559 T^{4} + 248 p T^{5} + 154 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 116 T^{2} - 392 T^{3} + 5158 T^{4} - 392 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 248 T^{2} - 36 p T^{3} + 20622 T^{4} - 36 p^{2} T^{5} + 248 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 266 T^{2} - 2136 T^{3} + 24423 T^{4} - 2136 p T^{5} + 266 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 502 T^{2} - 6088 T^{3} + 55063 T^{4} - 6088 p T^{5} + 502 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 30 T + 598 T^{2} + 7608 T^{3} + 73923 T^{4} + 7608 p T^{5} + 598 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 74 T^{2} - 424 T^{3} + 10903 T^{4} - 424 p T^{5} + 74 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 8 T + 208 T^{2} - 920 T^{3} + 17998 T^{4} - 920 p T^{5} + 208 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 21022 T^{4} + 1256 p T^{5} + 184 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 308 T^{2} + 2700 T^{3} + 37158 T^{4} + 2700 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 122 T^{2} + 1056 T^{3} + 14727 T^{4} + 1056 p T^{5} + 122 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 298 T^{2} + 956 T^{3} + 38551 T^{4} + 956 p T^{5} + 298 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.95650588509853899243655960850, −5.55153982518099371822449883205, −5.53587662728401401127953027312, −5.38010618761966940497442970981, −5.17774697704448126170539498316, −4.93148488746834441085294816448, −4.54705897694444845142250272921, −4.52499150314783164031021183604, −4.33782133692622593527616771527, −4.07590574891540434307633999927, −3.93307161307779342757834079421, −3.91375038412828249444600803438, −3.86745679446285289143281458663, −3.47528257782958138650524801517, −3.31251905765708186129095668421, −3.27942513046018057843345005413, −2.99598935602772914447834642343, −2.69418708577891787481258806412, −2.54311167586000211594567115505, −2.28756446685875360645255835729, −2.21494868329170422135866393037, −1.76592822037453618859097847637, −1.16615682181981048647370665586, −0.999571685834608483610930513553, −0.790935381588120258118418297716, 0, 0, 0, 0,
0.790935381588120258118418297716, 0.999571685834608483610930513553, 1.16615682181981048647370665586, 1.76592822037453618859097847637, 2.21494868329170422135866393037, 2.28756446685875360645255835729, 2.54311167586000211594567115505, 2.69418708577891787481258806412, 2.99598935602772914447834642343, 3.27942513046018057843345005413, 3.31251905765708186129095668421, 3.47528257782958138650524801517, 3.86745679446285289143281458663, 3.91375038412828249444600803438, 3.93307161307779342757834079421, 4.07590574891540434307633999927, 4.33782133692622593527616771527, 4.52499150314783164031021183604, 4.54705897694444845142250272921, 4.93148488746834441085294816448, 5.17774697704448126170539498316, 5.38010618761966940497442970981, 5.53587662728401401127953027312, 5.55153982518099371822449883205, 5.95650588509853899243655960850
