| L(s) = 1 | + 4·3-s − 12·17-s + 12·23-s − 4·25-s − 24·27-s − 20·43-s + 32·49-s − 48·51-s + 24·53-s + 8·61-s + 48·69-s − 16·75-s − 16·79-s − 42·81-s + 24·101-s + 40·103-s − 12·107-s − 36·113-s + 76·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 128·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 2.91·17-s + 2.50·23-s − 4/5·25-s − 4.61·27-s − 3.04·43-s + 32/7·49-s − 6.72·51-s + 3.29·53-s + 1.02·61-s + 5.77·69-s − 1.84·75-s − 1.80·79-s − 4.66·81-s + 2.38·101-s + 3.94·103-s − 1.16·107-s − 3.38·113-s + 6.90·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·147-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3001048410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3001048410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 \) |
| good | 3 | \( ( 1 - 2 T + 2 p T^{2} - 8 T^{3} + 19 T^{4} - 8 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 - 32 T^{2} + 466 T^{4} - 4352 T^{6} + 32419 T^{8} - 4352 p^{2} T^{10} + 466 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 6 T + 50 T^{2} + 252 T^{3} + 1095 T^{4} + 252 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 46 T^{2} + 1059 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + 74 T^{2} - 384 T^{3} + 2343 T^{4} - 384 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 74 T^{2} - 96 T^{3} + 2571 T^{4} - 96 p T^{5} + 74 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 28 T^{2} + 1926 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 104 T^{2} + 5482 T^{4} - 286832 T^{6} + 13046203 T^{8} - 286832 p^{2} T^{10} + 5482 p^{4} T^{12} - 104 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 86 T^{2} + 4779 T^{4} - 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 106 T^{2} + 376 T^{3} + 3175 T^{4} + 376 p T^{5} + 106 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 12 T + 56 T^{2} + 12 T^{3} - 306 T^{4} + 12 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 244 T^{2} + 31714 T^{4} - 2814496 T^{6} + 187865899 T^{8} - 2814496 p^{2} T^{10} + 31714 p^{4} T^{12} - 244 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 4 T + 142 T^{2} - 88 T^{3} + 8911 T^{4} - 88 p T^{5} + 142 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 200 T^{2} + 21898 T^{4} - 1932080 T^{6} + 143385115 T^{8} - 1932080 p^{2} T^{10} + 21898 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( 1 - 100 T^{2} + 14194 T^{4} - 1228768 T^{6} + 97338715 T^{8} - 1228768 p^{2} T^{10} + 14194 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 320 T^{2} + 54748 T^{4} - 6368960 T^{6} + 541047430 T^{8} - 6368960 p^{2} T^{10} + 54748 p^{4} T^{12} - 320 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 8 T + 136 T^{2} + 392 T^{3} + 8638 T^{4} + 392 p T^{5} + 136 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 184 T^{2} + 17788 T^{4} - 2042440 T^{6} + 209794726 T^{8} - 2042440 p^{2} T^{10} + 17788 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 - 484 T^{2} + 109810 T^{4} - 15753184 T^{6} + 1621395931 T^{8} - 15753184 p^{2} T^{10} + 109810 p^{4} T^{12} - 484 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( 1 - 368 T^{2} + 75346 T^{4} - 10665824 T^{6} + 1158959203 T^{8} - 10665824 p^{2} T^{10} + 75346 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−3.46743128771482986917936382382, −3.38361694125636597608680694803, −3.32590329624150890591192653227, −3.28954931493021210100149798914, −3.03584033439023002778487726221, −2.96587880813309476487503032959, −2.87100313799707752753737409429, −2.68804658863781236945279392087, −2.49387704102364303022550732024, −2.44336249274355565043184318982, −2.39111294339031115176473140353, −2.27838989460491709124292579070, −2.23353499342802024713282035244, −2.16404612762182486822507371942, −2.09906875004037978981109085270, −1.82120365429999100370311423133, −1.70166289359425860790954991561, −1.32792801023467508076199930019, −1.25651092212177882352233756670, −1.06421093969370159514749891295, −1.05086459116767829827626299732, −0.816829425341699260314534919163, −0.59516300462321637334736266986, −0.15436777573889010360149021476, −0.06902947732918989446907537546,
0.06902947732918989446907537546, 0.15436777573889010360149021476, 0.59516300462321637334736266986, 0.816829425341699260314534919163, 1.05086459116767829827626299732, 1.06421093969370159514749891295, 1.25651092212177882352233756670, 1.32792801023467508076199930019, 1.70166289359425860790954991561, 1.82120365429999100370311423133, 2.09906875004037978981109085270, 2.16404612762182486822507371942, 2.23353499342802024713282035244, 2.27838989460491709124292579070, 2.39111294339031115176473140353, 2.44336249274355565043184318982, 2.49387704102364303022550732024, 2.68804658863781236945279392087, 2.87100313799707752753737409429, 2.96587880813309476487503032959, 3.03584033439023002778487726221, 3.28954931493021210100149798914, 3.32590329624150890591192653227, 3.38361694125636597608680694803, 3.46743128771482986917936382382
Plot not available for L-functions of degree greater than 10.
