| L(s) = 1 | + 32·7-s − 2.91e3·13-s − 2.29e4·19-s + 4.56e4·25-s − 4.74e4·31-s − 1.22e5·37-s − 2.91e4·43-s − 9.21e4·49-s − 4.05e5·61-s − 8.88e5·67-s + 4.14e5·73-s − 2.61e6·79-s − 9.31e4·91-s − 1.64e6·97-s − 5.33e6·103-s + 2.40e6·109-s + 5.24e6·121-s + ⋯ |
| L(s) = 1 | + 0.0932·7-s − 1.32·13-s − 3.34·19-s + 2.91·25-s − 1.59·31-s − 2.41·37-s − 0.366·43-s − 0.782·49-s − 1.78·61-s − 2.95·67-s + 1.06·73-s − 5.30·79-s − 0.123·91-s − 1.80·97-s − 4.87·103-s + 1.85·109-s + 2.96·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.538419837\times10^{-5}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.538419837\times10^{-5}\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 912 p^{2} T^{2} + p^{12} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 16 T + 46434 T^{2} - 16 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 5248932 T^{2} + 12347300439398 T^{4} - 5248932 p^{12} T^{6} + p^{24} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 112 p T - 1907790 T^{2} + 112 p^{7} T^{3} + p^{12} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 48178624 T^{2} + 1745245170977154 T^{4} - 48178624 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 11456 T + 99696114 T^{2} + 11456 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 292713412 T^{2} + 43318920207311430 T^{4} - 292713412 p^{12} T^{6} + p^{24} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2211112800 T^{2} + 1924089603524685794 T^{4} - 2211112800 p^{12} T^{6} + p^{24} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 23728 T + 1385063106 T^{2} + 23728 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 61204 T + 4024489974 T^{2} + 61204 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 7658071872 T^{2} + 33889347787398181058 T^{4} - 7658071872 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 14560 T + 11546286546 T^{2} + 14560 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4689563268 T^{2} - 87222347054153806714 T^{4} - 4689563268 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 73413880800 T^{2} + 808875196296049586 p^{2} T^{4} - 73413880800 p^{12} T^{6} + p^{24} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 77019124260 T^{2} + \)\(46\!\cdots\!94\)\( T^{4} - 77019124260 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 202956 T + 62252402006 T^{2} + 202956 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 444064 T + 176341020594 T^{2} + 444064 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11781889860 T^{2} + \)\(20\!\cdots\!14\)\( T^{4} - 11781889860 p^{12} T^{6} + p^{24} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 207168 T + 305224316642 T^{2} - 207168 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 1308112 T + 904736729730 T^{2} + 1308112 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 447388429540 T^{2} + \)\(18\!\cdots\!34\)\( T^{4} - 447388429540 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1069485383040 T^{2} + \)\(69\!\cdots\!42\)\( T^{4} - 1069485383040 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 823264 T + 1631040388482 T^{2} + 823264 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
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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
−7.39463850069849935976302753386, −7.29101152367397203468749103504, −7.13316339954794843261740751362, −6.71070657637646542980305761775, −6.43692843879097212639577444611, −6.25823887633537076382629242822, −6.08573695059320250685874229707, −5.63906570914460013510666622118, −5.16999553189120210037486536482, −5.03732060440705806542067566326, −4.96890649114908923935616026927, −4.50304987109614198962203365982, −4.28397341380010311764627539579, −3.90770325853583457575543980845, −3.81504465152181829149494726405, −3.19984761289410313096176103483, −2.82454233254660057887528069909, −2.69125817225778627068793902865, −2.51803694795829272568764299443, −1.86737739085541444168599767515, −1.65638577585257317005247880615, −1.34903086571321468617452893569, −1.14953743437758647049304425757, −0.02797909427646163362682141270, −0.01269208549561431477275451800,
0.01269208549561431477275451800, 0.02797909427646163362682141270, 1.14953743437758647049304425757, 1.34903086571321468617452893569, 1.65638577585257317005247880615, 1.86737739085541444168599767515, 2.51803694795829272568764299443, 2.69125817225778627068793902865, 2.82454233254660057887528069909, 3.19984761289410313096176103483, 3.81504465152181829149494726405, 3.90770325853583457575543980845, 4.28397341380010311764627539579, 4.50304987109614198962203365982, 4.96890649114908923935616026927, 5.03732060440705806542067566326, 5.16999553189120210037486536482, 5.63906570914460013510666622118, 6.08573695059320250685874229707, 6.25823887633537076382629242822, 6.43692843879097212639577444611, 6.71070657637646542980305761775, 7.13316339954794843261740751362, 7.29101152367397203468749103504, 7.39463850069849935976302753386
