| L(s) = 1 | − 8·2-s + 48·4-s − 50·5-s + 169·7-s − 256·8-s + 400·10-s − 327·11-s + 634·13-s − 1.35e3·14-s + 1.28e3·16-s + 303·17-s − 353·19-s − 2.40e3·20-s + 2.61e3·22-s − 1.04e3·23-s + 1.87e3·25-s − 5.07e3·26-s + 8.11e3·28-s − 4.00e3·29-s + 2.77e3·31-s − 6.14e3·32-s − 2.42e3·34-s − 8.45e3·35-s + 1.07e3·37-s + 2.82e3·38-s + 1.28e4·40-s − 3.99e3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.30·7-s − 1.41·8-s + 1.26·10-s − 0.814·11-s + 1.04·13-s − 1.84·14-s + 5/4·16-s + 0.254·17-s − 0.224·19-s − 1.34·20-s + 1.15·22-s − 0.410·23-s + 3/5·25-s − 1.47·26-s + 1.95·28-s − 0.884·29-s + 0.518·31-s − 1.06·32-s − 0.359·34-s − 1.16·35-s + 0.129·37-s + 0.317·38-s + 1.26·40-s − 0.370·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72900 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.541148291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541148291\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 169 T + 18972 T^{2} - 169 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 327 T + 29732 p T^{2} + 327 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 634 T + 38043 p T^{2} - 634 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 303 T - 818534 T^{2} - 303 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 353 T + 1302150 T^{2} + 353 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1041 T + 13121824 T^{2} + 1041 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4005 T + 42396652 T^{2} + 4005 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2773 T + 59158902 T^{2} - 2773 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 1075 T + 138780780 T^{2} - 1075 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3990 T + 197268538 T^{2} + 3990 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7621 T + 271920834 T^{2} - 7621 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12801 T + 499111858 T^{2} + 12801 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7482 T + 835661266 T^{2} - 7482 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 29436 T + 1060612726 T^{2} + 29436 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 54073 T + 2387034132 T^{2} - 54073 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14173 T + 1334077890 T^{2} - 14173 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 71334 T + 2677885342 T^{2} - 71334 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 42565 T + 4128395352 T^{2} - 42565 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 166258 T + 12986127339 T^{2} - 166258 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 133992 T + 11722139218 T^{2} - 133992 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 92556 T + 8538936658 T^{2} - 92556 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 193441 T + 18869829204 T^{2} - 193441 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05446700775271737640204025814, −10.97379903578696867944666027716, −10.36563676781935142263446877863, −9.987622032229201575090183215570, −9.206017493877693262872505710424, −8.882381159046499083542752785571, −8.213601637912817895445168419173, −8.055937324644147205358656149042, −7.68402540080225293382629231397, −7.22767421345294684045120456987, −6.36722354155671497855349139960, −6.09304375144746370787950252238, −4.98509635324684641388802291924, −4.93684973776759580925853325974, −3.65308502191252149468231672162, −3.52424534871388476386676139249, −2.23154601152061194875796747250, −1.97945457098692327494116249562, −0.893700122686386241577461447462, −0.58180314314507464397337456442,
0.58180314314507464397337456442, 0.893700122686386241577461447462, 1.97945457098692327494116249562, 2.23154601152061194875796747250, 3.52424534871388476386676139249, 3.65308502191252149468231672162, 4.93684973776759580925853325974, 4.98509635324684641388802291924, 6.09304375144746370787950252238, 6.36722354155671497855349139960, 7.22767421345294684045120456987, 7.68402540080225293382629231397, 8.055937324644147205358656149042, 8.213601637912817895445168419173, 8.882381159046499083542752785571, 9.206017493877693262872505710424, 9.987622032229201575090183215570, 10.36563676781935142263446877863, 10.97379903578696867944666027716, 11.05446700775271737640204025814
