| L(s) = 1 | − 16·7-s + 8·11-s + 16·17-s − 48·23-s − 56·31-s + 32·37-s − 8·41-s + 128·43-s + 80·47-s + 128·49-s − 32·53-s − 120·61-s − 96·67-s + 32·71-s − 256·73-s − 128·77-s − 9·81-s − 160·83-s + 160·97-s + 464·101-s + 336·103-s + 96·107-s + 400·113-s − 256·119-s − 252·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 8/11·11-s + 0.941·17-s − 2.08·23-s − 1.80·31-s + 0.864·37-s − 0.195·41-s + 2.97·43-s + 1.70·47-s + 2.61·49-s − 0.603·53-s − 1.96·61-s − 1.43·67-s + 0.450·71-s − 3.50·73-s − 1.66·77-s − 1/9·81-s − 1.92·83-s + 1.64·97-s + 4.59·101-s + 3.26·103-s + 0.897·107-s + 3.53·113-s − 2.15·119-s − 2.08·121-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.097978397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.097978397\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1104 T^{3} + 9122 T^{4} + 1104 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 150 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 49154 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 4368 T^{3} + 148802 T^{4} - 4368 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 284 T^{2} - 20250 T^{4} - 284 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 48 T + 1152 T^{2} + 30000 T^{3} + 772034 T^{4} + 30000 p^{2} T^{5} + 1152 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2084 T^{2} + 2107110 T^{4} - 2084 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 28 T + 582 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 29088 T^{3} + 1440962 T^{4} - 29088 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T - 90 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 128 T + 8192 T^{2} - 474240 T^{3} + 24009314 T^{4} - 474240 p^{2} T^{5} + 8192 p^{4} T^{6} - 128 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T + 3200 T^{2} - 144720 T^{3} + 6384962 T^{4} - 144720 p^{2} T^{5} + 3200 p^{4} T^{6} - 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 32 T + 512 T^{2} - 30432 T^{3} - 12328798 T^{4} - 30432 p^{2} T^{5} + 512 p^{4} T^{6} + 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6236 T^{2} + 20094246 T^{4} - 6236 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 60 T + 2198 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 96 T + 4608 T^{2} - 16032 T^{3} - 21622558 T^{4} - 16032 p^{2} T^{5} + 4608 p^{4} T^{6} + 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 6690 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 256 T + 32768 T^{2} + 3350784 T^{3} + 282426242 T^{4} + 3350784 p^{2} T^{5} + 32768 p^{4} T^{6} + 256 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 8126 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 160 T + 12800 T^{2} + 992160 T^{3} + 76431458 T^{4} + 992160 p^{2} T^{5} + 12800 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 30716 T^{2} + 361199046 T^{4} - 30716 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 160 T + 12800 T^{2} + 755040 T^{3} - 155062462 T^{4} + 755040 p^{2} T^{5} + 12800 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.83990735533077893529012539374, −6.29865778602870207940649679469, −6.09921271549062566248059731974, −6.03336595576211551181073267974, −6.01693033156996802249736205811, −5.96220422357060009234558304266, −5.53922475403133579530070022045, −5.47325326331912976793103726110, −4.69444696743113104750385526780, −4.67518891433300384565794082842, −4.44159227505069449618913390317, −4.32669412394470304656762295937, −3.78173447327228956325290912014, −3.71954309282483221143180889567, −3.55833140231100691265084448091, −3.18801950096158660486568416883, −2.94061161462417043408564306617, −2.89132341783795606850972403902, −2.29644299863557770380018414974, −2.13784420043216426377822309159, −1.68927812641132156456619738160, −1.56073479782314628437831663901, −0.72008924969337457855181175102, −0.70482936395277045562395804632, −0.28016925355413153369584290186,
0.28016925355413153369584290186, 0.70482936395277045562395804632, 0.72008924969337457855181175102, 1.56073479782314628437831663901, 1.68927812641132156456619738160, 2.13784420043216426377822309159, 2.29644299863557770380018414974, 2.89132341783795606850972403902, 2.94061161462417043408564306617, 3.18801950096158660486568416883, 3.55833140231100691265084448091, 3.71954309282483221143180889567, 3.78173447327228956325290912014, 4.32669412394470304656762295937, 4.44159227505069449618913390317, 4.67518891433300384565794082842, 4.69444696743113104750385526780, 5.47325326331912976793103726110, 5.53922475403133579530070022045, 5.96220422357060009234558304266, 6.01693033156996802249736205811, 6.03336595576211551181073267974, 6.09921271549062566248059731974, 6.29865778602870207940649679469, 6.83990735533077893529012539374
