L(s) = 1 | + 4·3-s + 4·4-s − 10·7-s + 3·9-s + 16·12-s − 10·13-s + 14·16-s − 10·19-s − 40·21-s − 14·25-s − 10·27-s − 40·28-s + 10·31-s + 12·36-s − 6·37-s − 40·39-s + 56·48-s + 31·49-s − 40·52-s − 40·57-s − 20·61-s − 30·63-s + 35·64-s − 4·67-s − 80·73-s − 56·75-s − 40·76-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 2·4-s − 3.77·7-s + 9-s + 4.61·12-s − 2.77·13-s + 7/2·16-s − 2.29·19-s − 8.72·21-s − 2.79·25-s − 1.92·27-s − 7.55·28-s + 1.79·31-s + 2·36-s − 0.986·37-s − 6.40·39-s + 8.08·48-s + 31/7·49-s − 5.54·52-s − 5.29·57-s − 2.56·61-s − 3.77·63-s + 35/8·64-s − 0.488·67-s − 9.36·73-s − 6.46·75-s − 4.58·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3377720842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3377720842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 4 T + 13 T^{2} - 10 p T^{3} + 61 T^{4} - 10 p^{2} T^{5} + 13 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p^{2} T^{2} + p T^{4} + 13 T^{6} - 35 T^{8} + 13 p^{2} T^{10} + p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + 14 T^{2} + 51 T^{4} - 356 T^{6} - 3739 T^{8} - 356 p^{2} T^{10} + 51 p^{4} T^{12} + 14 p^{6} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 5 T + 22 T^{2} + 85 T^{3} + 229 T^{4} + 85 p T^{5} + 22 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 5 T + 28 T^{2} + 145 T^{3} + 499 T^{4} + 145 p T^{5} + 28 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 9 T^{2} + 267 T^{4} - 131 p T^{6} + 32880 T^{8} - 131 p^{3} T^{10} + 267 p^{4} T^{12} - 9 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 5 T + p T^{2} - 5 p T^{3} - 464 T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 50 T^{2} + 1363 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 38 T^{2} + 1523 T^{4} - 1784 p T^{6} + 935125 T^{8} - 1784 p^{3} T^{10} + 1523 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 5 T + 9 T^{2} - 205 T^{3} + 1916 T^{4} - 205 p T^{5} + 9 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 67 T^{2} + 4308 T^{4} - 255209 T^{6} + 10464455 T^{8} - 255209 p^{2} T^{10} + 4308 p^{4} T^{12} - 67 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 122 T^{2} + 6919 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 50 T^{2} - 969 T^{4} - 128660 T^{6} - 3507499 T^{8} - 128660 p^{2} T^{10} - 969 p^{4} T^{12} + 50 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 115 T^{2} + 6051 T^{4} + 72185 T^{6} - 4134064 T^{8} + 72185 p^{2} T^{10} + 6051 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 107 T^{2} + 9663 T^{4} + 800929 T^{6} + 52808600 T^{8} + 800929 p^{2} T^{10} + 9663 p^{4} T^{12} + 107 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 10 T + 111 T^{2} + 1040 T^{3} + 8421 T^{4} + 1040 p T^{5} + 111 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 + 162 T^{2} + 5103 T^{4} - 803356 T^{6} - 100375395 T^{8} - 803356 p^{2} T^{10} + 5103 p^{4} T^{12} + 162 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( ( 1 + 40 T + 713 T^{2} + 8040 T^{3} + 72529 T^{4} + 8040 p T^{5} + 713 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 5 T + 114 T^{2} - 1705 T^{3} + 6861 T^{4} - 1705 p T^{5} + 114 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 231 T^{2} + 16452 T^{4} - 260413 T^{6} - 14194425 T^{8} - 260413 p^{2} T^{10} + 16452 p^{4} T^{12} - 231 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 266 T^{2} + 31531 T^{4} - 266 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 8 T - 63 T^{2} + 980 T^{3} - 1099 T^{4} + 980 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.00596402827226760492594864492, −4.93016619105133314839667333584, −4.61508540997712200367672792801, −4.60108521860180637569954720900, −4.45434976104275683635833167015, −4.33692816221047858844447637049, −4.05381155804290206205801057155, −4.02022508995039669905678145578, −3.74692529353032790963007090986, −3.39635023676419577632719303102, −3.31182478484777482109723416886, −3.17136165296563925556564549555, −3.12298501473450541086909625309, −3.07030284080205704574213372561, −2.94854753321700163716359743653, −2.82549936478896593049525913961, −2.76896069087677943810417635973, −2.18362632536400690894373680815, −2.15695794138535898997767186750, −2.10726026587560049406927318552, −2.02504153201731218128099537840, −1.60729494305642463209203113221, −1.50003477949630856131553217432, −0.54258761289188424168772325784, −0.12303124040700813944607700699,
0.12303124040700813944607700699, 0.54258761289188424168772325784, 1.50003477949630856131553217432, 1.60729494305642463209203113221, 2.02504153201731218128099537840, 2.10726026587560049406927318552, 2.15695794138535898997767186750, 2.18362632536400690894373680815, 2.76896069087677943810417635973, 2.82549936478896593049525913961, 2.94854753321700163716359743653, 3.07030284080205704574213372561, 3.12298501473450541086909625309, 3.17136165296563925556564549555, 3.31182478484777482109723416886, 3.39635023676419577632719303102, 3.74692529353032790963007090986, 4.02022508995039669905678145578, 4.05381155804290206205801057155, 4.33692816221047858844447637049, 4.45434976104275683635833167015, 4.60108521860180637569954720900, 4.61508540997712200367672792801, 4.93016619105133314839667333584, 5.00596402827226760492594864492
Plot not available for L-functions of degree greater than 10.