| L(s) = 1 | + 2·3-s − 9-s + 4·13-s − 20·25-s − 2·27-s − 20·31-s + 8·39-s + 12·49-s + 4·73-s − 40·75-s − 7·81-s − 40·93-s − 4·117-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 24·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 78·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/3·9-s + 1.10·13-s − 4·25-s − 0.384·27-s − 3.59·31-s + 1.28·39-s + 12/7·49-s + 0.468·73-s − 4.61·75-s − 7/9·81-s − 4.14·93-s − 0.369·117-s − 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 6·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5050096459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5050096459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( 1 - 20 T^{2} + 70 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| good | 5 | \( ( 1 + 2 p T^{2} + 58 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 6 T^{2} + 90 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 10 T^{2} - 230 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 6 T^{2} + 714 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 57 T^{2} + 2456 T^{4} - 57 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 56 T^{2} + 1822 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 49 T^{2} + 864 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 90 T^{2} + 5706 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 65 T^{2} + 1900 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 22 T^{2} + 4906 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 176 T^{2} + 14638 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 156 T^{2} + 13254 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 70 T^{2} + 8826 T^{4} - 70 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 269 T^{2} + 28168 T^{4} - 269 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 108 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 170 T^{2} + 14794 T^{4} - 170 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 216 T^{2} + 22110 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 294 T^{2} + 36618 T^{4} + 294 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 272 T^{2} + 35614 T^{4} - 272 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−5.41283402004393356135471671045, −5.31009214935167703773922651489, −5.07445497586325710217459304795, −4.99429887855410325458190111892, −4.69415630527137592386083491106, −4.54085846459526942227401791312, −4.26048010384683140462395635308, −4.16008693883498986071377394083, −4.05276797566720825448117518440, −3.84869294200729240636375506270, −3.80070058080213375694752293376, −3.66810252290702906778283339261, −3.43429768865754316565780590496, −3.28246583349053593469215444275, −3.12225864317805090074164937788, −2.91571866804644427937028347328, −2.65496413717944355497870004473, −2.46519556850254437040356775606, −2.17501141909324510879865115537, −2.09300117862288481132216940129, −1.82047210164072324471187938083, −1.81830820278618611233483936579, −1.20710062859298750325127459376, −1.15459944877248117764411251767, −0.16515014063382521903013245421,
0.16515014063382521903013245421, 1.15459944877248117764411251767, 1.20710062859298750325127459376, 1.81830820278618611233483936579, 1.82047210164072324471187938083, 2.09300117862288481132216940129, 2.17501141909324510879865115537, 2.46519556850254437040356775606, 2.65496413717944355497870004473, 2.91571866804644427937028347328, 3.12225864317805090074164937788, 3.28246583349053593469215444275, 3.43429768865754316565780590496, 3.66810252290702906778283339261, 3.80070058080213375694752293376, 3.84869294200729240636375506270, 4.05276797566720825448117518440, 4.16008693883498986071377394083, 4.26048010384683140462395635308, 4.54085846459526942227401791312, 4.69415630527137592386083491106, 4.99429887855410325458190111892, 5.07445497586325710217459304795, 5.31009214935167703773922651489, 5.41283402004393356135471671045
Plot not available for L-functions of degree greater than 10.
