| L(s) = 1 | + 12·3-s − 8·4-s + 90·9-s − 96·12-s + 48·16-s + 24·17-s − 348·23-s + 422·25-s + 540·27-s − 564·29-s − 720·36-s − 748·43-s + 576·48-s + 1.06e3·49-s + 288·51-s − 348·53-s − 8·61-s − 256·64-s − 192·68-s − 4.17e3·69-s + 5.06e3·75-s − 920·79-s + 2.83e3·81-s − 6.76e3·87-s + 2.78e3·92-s − 3.37e3·100-s − 1.59e3·101-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 3/4·16-s + 0.342·17-s − 3.15·23-s + 3.37·25-s + 3.84·27-s − 3.61·29-s − 3.33·36-s − 2.65·43-s + 1.73·48-s + 3.09·49-s + 0.790·51-s − 0.901·53-s − 0.0167·61-s − 1/2·64-s − 0.342·68-s − 7.28·69-s + 7.79·75-s − 1.31·79-s + 35/9·81-s − 8.34·87-s + 3.15·92-s − 3.37·100-s − 1.57·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.018595878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018595878\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 422 T^{2} + 75339 T^{4} - 422 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1060 T^{2} + 510906 T^{4} - 1060 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2852 T^{2} + 4098186 T^{4} - 2852 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 8539 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1204 T^{2} - 73486806 T^{4} - 1204 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 174 T + 1292 p T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 282 T + 67687 T^{2} + 282 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 112156 T^{2} + 4912891014 T^{4} - 112156 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 43618 T^{2} - 85335501 T^{4} - 43618 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 216962 T^{2} + 20865212451 T^{4} - 216962 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 374 T + 184236 T^{2} + 374 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 153044 T^{2} + 25540546842 T^{4} - 153044 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 174 T + 300031 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 709388 T^{2} + 210041056566 T^{4} - 709388 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 4 T + 249603 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 910564 T^{2} + 387569626410 T^{4} - 910564 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20228 T^{2} - 106395526470 T^{4} - 20228 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 448846 T^{2} + 314911329555 T^{4} - 448846 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 460 T + 524790 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1332116 T^{2} + 901584332010 T^{4} - 1332116 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1012292 T^{2} + 484537982310 T^{4} - 1012292 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2727868 T^{2} + 3492562141446 T^{4} - 2727868 p^{6} T^{6} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.76341297749319466544640946006, −6.64864088731203822169028473256, −6.32725296009014335992749449066, −6.01898342639063173903153717831, −5.76603604483539836722724955008, −5.59790421906716510569683341020, −5.33172197782347834873370403246, −4.95541351739381782011971916616, −4.88548233604556586605737093591, −4.52563327052207436836721671268, −4.27473958587787815659378006382, −4.01673028814467443518947115931, −3.81924304019032013511648190989, −3.65295131774191246961486239530, −3.32124656293430065240362892731, −3.30709470795912883518520747017, −2.73230000792478594799829163630, −2.64191502132425859971526706994, −2.25443890784242459245028586753, −1.99322722065048801605128640126, −1.61664528629271223503015000430, −1.50552590034572440120071896637, −1.08126117766784415950865424886, −0.58634915467317289288639000543, −0.13917715477113927400736780826,
0.13917715477113927400736780826, 0.58634915467317289288639000543, 1.08126117766784415950865424886, 1.50552590034572440120071896637, 1.61664528629271223503015000430, 1.99322722065048801605128640126, 2.25443890784242459245028586753, 2.64191502132425859971526706994, 2.73230000792478594799829163630, 3.30709470795912883518520747017, 3.32124656293430065240362892731, 3.65295131774191246961486239530, 3.81924304019032013511648190989, 4.01673028814467443518947115931, 4.27473958587787815659378006382, 4.52563327052207436836721671268, 4.88548233604556586605737093591, 4.95541351739381782011971916616, 5.33172197782347834873370403246, 5.59790421906716510569683341020, 5.76603604483539836722724955008, 6.01898342639063173903153717831, 6.32725296009014335992749449066, 6.64864088731203822169028473256, 6.76341297749319466544640946006
