| L(s) = 1 | − 8·2-s − 4·3-s + 40·4-s + 32·6-s + 4·7-s − 160·8-s + 9·9-s − 160·12-s − 32·14-s + 560·16-s − 72·18-s − 16·21-s − 44·23-s + 640·24-s + 160·28-s + 22·29-s − 1.79e3·32-s + 360·36-s − 62·41-s + 128·42-s + 76·43-s + 352·46-s + 4·47-s − 2.24e3·48-s + 49·49-s − 640·56-s − 176·58-s + ⋯ |
| L(s) = 1 | − 4·2-s − 4/3·3-s + 10·4-s + 16/3·6-s + 4/7·7-s − 20·8-s + 9-s − 13.3·12-s − 2.28·14-s + 35·16-s − 4·18-s − 0.761·21-s − 1.91·23-s + 80/3·24-s + 40/7·28-s + 0.758·29-s − 56·32-s + 10·36-s − 1.51·41-s + 3.04·42-s + 1.76·43-s + 7.65·46-s + 4/47·47-s − 46.6·48-s + 49-s − 11.4·56-s − 3.03·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4645602673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4645602673\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 4 T + 7 T^{2} - 8 T^{3} - 95 T^{4} - 8 p^{2} T^{5} + 7 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T - 33 T^{2} + 328 T^{3} + 305 T^{4} + 328 p^{2} T^{5} - 33 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_4\times C_2$ | \( 1 + 44 T + 1407 T^{2} + 38632 T^{3} + 955505 T^{4} + 38632 p^{2} T^{5} + 1407 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_4\times C_2$ | \( 1 - 22 T - 357 T^{2} + 26356 T^{3} - 279595 T^{4} + 26356 p^{2} T^{5} - 357 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 + 62 T + 2163 T^{2} + 29884 T^{3} - 1783195 T^{4} + 29884 p^{2} T^{5} + 2163 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_4\times C_2$ | \( 1 - 76 T + 3927 T^{2} - 157928 T^{3} + 4741505 T^{4} - 157928 p^{2} T^{5} + 3927 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_4\times C_2$ | \( 1 - 4 T - 2193 T^{2} + 17608 T^{3} + 4773905 T^{4} + 17608 p^{2} T^{5} - 2193 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_4\times C_2$ | \( 1 - 58 T - 357 T^{2} + 236524 T^{3} - 12389995 T^{4} + 236524 p^{2} T^{5} - 357 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 116 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_4\times C_2$ | \( 1 - 76 T - 1113 T^{2} + 608152 T^{3} - 38552095 T^{4} + 608152 p^{2} T^{5} - 1113 p^{4} T^{6} - 76 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 142 T + 12243 T^{2} - 613724 T^{3} - 9827995 T^{4} - 613724 p^{2} T^{5} + 12243 p^{4} T^{6} - 142 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 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
−7.79172649585372019361157197359, −7.68499499248913521169729001547, −7.18407951846086246401222863502, −7.07102539945270009865457307526, −6.83620114144393235080445824818, −6.56753597426655459055960330274, −6.49766071095867317497696142480, −6.12043063769721288042047763573, −6.05697292759037617477110707768, −5.52815982365851018727476250934, −5.40277763117106705015527683874, −5.32413800878852874627091335560, −4.81794418392175809054611087502, −4.30318860135218079893840895064, −3.79213044867415041194263828010, −3.53836217862552694959734318761, −3.40198270030470537274487977817, −2.67475431640778865200818475787, −2.30927290119245453405655482387, −2.27430748935110502883375386734, −1.85523651777784252991166294489, −1.52291576488252878829884131357, −0.802585640563866415878460231872, −0.59767780193276116918191958778, −0.58474292221160838027086649990,
0.58474292221160838027086649990, 0.59767780193276116918191958778, 0.802585640563866415878460231872, 1.52291576488252878829884131357, 1.85523651777784252991166294489, 2.27430748935110502883375386734, 2.30927290119245453405655482387, 2.67475431640778865200818475787, 3.40198270030470537274487977817, 3.53836217862552694959734318761, 3.79213044867415041194263828010, 4.30318860135218079893840895064, 4.81794418392175809054611087502, 5.32413800878852874627091335560, 5.40277763117106705015527683874, 5.52815982365851018727476250934, 6.05697292759037617477110707768, 6.12043063769721288042047763573, 6.49766071095867317497696142480, 6.56753597426655459055960330274, 6.83620114144393235080445824818, 7.07102539945270009865457307526, 7.18407951846086246401222863502, 7.68499499248913521169729001547, 7.79172649585372019361157197359
