| L(s) = 1 | + 2·2-s − 2·3-s − 10·4-s + 2·5-s − 4·6-s + 20·7-s − 32·8-s − 3·9-s + 4·10-s − 22·11-s + 20·12-s + 80·13-s + 40·14-s − 4·15-s + 44·16-s − 124·17-s − 6·18-s + 72·19-s − 20·20-s − 40·21-s − 44·22-s − 98·23-s + 64·24-s − 55·25-s + 160·26-s − 34·27-s − 200·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.384·3-s − 5/4·4-s + 0.178·5-s − 0.272·6-s + 1.07·7-s − 1.41·8-s − 1/9·9-s + 0.126·10-s − 0.603·11-s + 0.481·12-s + 1.70·13-s + 0.763·14-s − 0.0688·15-s + 0.687·16-s − 1.76·17-s − 0.0785·18-s + 0.869·19-s − 0.223·20-s − 0.415·21-s − 0.426·22-s − 0.888·23-s + 0.544·24-s − 0.439·25-s + 1.20·26-s − 0.242·27-s − 1.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8380992720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8380992720\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 7 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 59 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 20 T + 738 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 80 T + 4794 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13238 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 72 T + 4214 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 98 T + 22847 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 44554 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 34 T + 57519 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 54 T + 101843 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 536 T + 209618 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 60 T + 159146 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 272 T + 182942 T^{2} + 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 492 T + 348862 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 634 T + 458975 T^{2} - 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 840 T + 528794 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 754 T + 742455 T^{2} - 754 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 678 T + 813415 T^{2} + 678 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 400 T + 160962 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 p T - 279966 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 468 T + 1155130 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1842 T + 1935427 T^{2} + 1842 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2194 T + 2966547 T^{2} - 2194 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.86400651781384717306399183390, −19.98272888814166802692057074899, −18.95082872831269086040641176420, −18.07786919972305181705348371154, −17.74965087372963331749051957749, −17.67802746306553362637184316339, −16.06159423926900669861311921151, −15.70426635999610925768034563766, −14.45413212802334498598783244085, −14.09492608114868723018686830721, −13.23394761241722868999602498309, −12.94543933526161124466296837781, −11.54019096455172857387002928910, −11.08380472054021309562065233191, −9.795879747848764729591001837096, −8.738362642763152216189473172240, −8.090000841979261224322862516068, −6.16039971848470745442367419454, −5.12519565114525393474177867591, −4.10903776582739696739959352561,
4.10903776582739696739959352561, 5.12519565114525393474177867591, 6.16039971848470745442367419454, 8.090000841979261224322862516068, 8.738362642763152216189473172240, 9.795879747848764729591001837096, 11.08380472054021309562065233191, 11.54019096455172857387002928910, 12.94543933526161124466296837781, 13.23394761241722868999602498309, 14.09492608114868723018686830721, 14.45413212802334498598783244085, 15.70426635999610925768034563766, 16.06159423926900669861311921151, 17.67802746306553362637184316339, 17.74965087372963331749051957749, 18.07786919972305181705348371154, 18.95082872831269086040641176420, 19.98272888814166802692057074899, 20.86400651781384717306399183390
