| 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
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(11.58283492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.58283492\) |
| \(L(4)\) |
|
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
−7.59769234868221333111316666356, −7.00395688400831941458315840976, −6.92125587039654304223984977260, −6.88923125471053443709949139951, −6.81813387935731106009471241088, −6.02112090700291356218366881746, −6.00506709651007202890757689360, −5.55876763210901684222063158171, −5.39965905176121934759138798128, −4.85670076996265374459130575716, −4.76185380623920537722984490075, −4.72884847363343861966961777084, −4.65136613529779505945269494317, −3.61943025758253518783518934668, −3.44453609377738089082776835726, −3.34000452240583408783563839389, −3.15672240497497653495279018412, −2.77488978802047181168702292464, −2.24132098618675330762834481182, −2.05139318455884037232143412426, −1.75567337179864569014625404976, −1.03369021926374190521567339229, −0.868392864913430845222172674994, −0.65438627870606956094206576438, −0.48013461765325271709458233091,
0.48013461765325271709458233091, 0.65438627870606956094206576438, 0.868392864913430845222172674994, 1.03369021926374190521567339229, 1.75567337179864569014625404976, 2.05139318455884037232143412426, 2.24132098618675330762834481182, 2.77488978802047181168702292464, 3.15672240497497653495279018412, 3.34000452240583408783563839389, 3.44453609377738089082776835726, 3.61943025758253518783518934668, 4.65136613529779505945269494317, 4.72884847363343861966961777084, 4.76185380623920537722984490075, 4.85670076996265374459130575716, 5.39965905176121934759138798128, 5.55876763210901684222063158171, 6.00506709651007202890757689360, 6.02112090700291356218366881746, 6.81813387935731106009471241088, 6.88923125471053443709949139951, 6.92125587039654304223984977260, 7.00395688400831941458315840976, 7.59769234868221333111316666356
