| L(s) = 1 | + 5·3-s − 2·4-s − 3·5-s + 7·7-s + 15·9-s − 10·12-s + 10·13-s − 15·15-s − 16-s − 5·17-s + 5·19-s + 6·20-s + 35·21-s − 2·23-s − 7·25-s + 35·27-s − 14·28-s + 3·29-s + 9·31-s − 2·32-s − 21·35-s − 30·36-s + 8·37-s + 50·39-s − 7·41-s + 43-s − 45·45-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 4-s − 1.34·5-s + 2.64·7-s + 5·9-s − 2.88·12-s + 2.77·13-s − 3.87·15-s − 1/4·16-s − 1.21·17-s + 1.14·19-s + 1.34·20-s + 7.63·21-s − 0.417·23-s − 7/5·25-s + 6.73·27-s − 2.64·28-s + 0.557·29-s + 1.61·31-s − 0.353·32-s − 3.54·35-s − 5·36-s + 1.31·37-s + 8.00·39-s − 1.09·41-s + 0.152·43-s − 6.70·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.728112360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.728112360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{5} \) |
| 67 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 + p T^{2} + 5 T^{4} + p T^{5} + 5 p T^{6} + p^{4} T^{8} + p^{5} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 + 3 T + 16 T^{2} + 41 T^{3} + p^{3} T^{4} + 276 T^{5} + p^{4} T^{6} + 41 p^{2} T^{7} + 16 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 - p T + 38 T^{2} - 19 p T^{3} + 425 T^{4} - 1112 T^{5} + 425 p T^{6} - 19 p^{3} T^{7} + 38 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 35 T^{2} - 4 T^{3} + 606 T^{4} - 120 T^{5} + 606 p T^{6} - 4 p^{2} T^{7} + 35 p^{3} T^{8} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 10 T + 85 T^{2} - 484 T^{3} + 2382 T^{4} - 9236 T^{5} + 2382 p T^{6} - 484 p^{2} T^{7} + 85 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 5 T + 39 T^{2} + 244 T^{3} + 1180 T^{4} + 4838 T^{5} + 1180 p T^{6} + 244 p^{2} T^{7} + 39 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 5 T + 49 T^{2} - 132 T^{3} + 808 T^{4} - 1422 T^{5} + 808 p T^{6} - 132 p^{2} T^{7} + 49 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 + 2 T + 101 T^{2} + 192 T^{3} + 4335 T^{4} + 6712 T^{5} + 4335 p T^{6} + 192 p^{2} T^{7} + 101 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 3 T + 47 T^{2} - 124 T^{3} + 1932 T^{4} - 4194 T^{5} + 1932 p T^{6} - 124 p^{2} T^{7} + 47 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 9 T + 154 T^{2} - 943 T^{3} + 9185 T^{4} - 41200 T^{5} + 9185 p T^{6} - 943 p^{2} T^{7} + 154 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 8 T + 117 T^{2} - 746 T^{3} + 6797 T^{4} - 34118 T^{5} + 6797 p T^{6} - 746 p^{2} T^{7} + 117 p^{3} T^{8} - 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 7 T + 190 T^{2} + 1019 T^{3} + 14951 T^{4} + 60056 T^{5} + 14951 p T^{6} + 1019 p^{2} T^{7} + 190 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - T + 124 T^{2} + 33 T^{3} + 8725 T^{4} + 480 T^{5} + 8725 p T^{6} + 33 p^{2} T^{7} + 124 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 + 5 T + 189 T^{2} + 692 T^{3} + 15424 T^{4} + 42974 T^{5} + 15424 p T^{6} + 692 p^{2} T^{7} + 189 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 15 T + 168 T^{2} + 1247 T^{3} + 8491 T^{4} + 46324 T^{5} + 8491 p T^{6} + 1247 p^{2} T^{7} + 168 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 6 T + 191 T^{2} + 1132 T^{3} + 18867 T^{4} + 91308 T^{5} + 18867 p T^{6} + 1132 p^{2} T^{7} + 191 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 6 T + 209 T^{2} - 420 T^{3} + 16170 T^{4} - 2732 T^{5} + 16170 p T^{6} - 420 p^{2} T^{7} + 209 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 22 T + 375 T^{2} - 4100 T^{3} + 42078 T^{4} - 349772 T^{5} + 42078 p T^{6} - 4100 p^{2} T^{7} + 375 p^{3} T^{8} - 22 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 81 T^{2} + 534 T^{3} + 11057 T^{4} - 874 T^{5} + 11057 p T^{6} + 534 p^{2} T^{7} + 81 p^{3} T^{8} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 28 T + 371 T^{2} - 2912 T^{3} + 17042 T^{4} - 111624 T^{5} + 17042 p T^{6} - 2912 p^{2} T^{7} + 371 p^{3} T^{8} - 28 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 9 T + 186 T^{2} - 169 T^{3} + 5585 T^{4} + 99852 T^{5} + 5585 p T^{6} - 169 p^{2} T^{7} + 186 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 11 T + 365 T^{2} + 3632 T^{3} + 59750 T^{4} + 469970 T^{5} + 59750 p T^{6} + 3632 p^{2} T^{7} + 365 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 14 T + 309 T^{2} + 1468 T^{3} + 20994 T^{4} - 15492 T^{5} + 20994 p T^{6} + 1468 p^{2} T^{7} + 309 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.010039276281686126806313404292, −7.87641497334521749919895850929, −7.67607952565961795904728471905, −7.55340183112394784194766656539, −6.82912986996465439482740748315, −6.77545329552074397306886529855, −6.72354865415525116019950534868, −6.26985799216267851454384098850, −6.08640589843678129026770434084, −5.71533954470351759201949442004, −5.17743806527440331177963645611, −4.96119971002619649983511471878, −4.84961506982248912812360369223, −4.56382031935202535692821598665, −4.42465799549446673700899685517, −3.85790963983727758183321413905, −3.83831486766162639447400563118, −3.73784388367155685194256090339, −3.62277151803108532434416368869, −2.89651627429618269030921966383, −2.72483366108877983658896412610, −2.33556550242256407561875745858, −1.76709656021757708569328754194, −1.46651719108754573388745052345, −1.28439704256392910340477817898,
1.28439704256392910340477817898, 1.46651719108754573388745052345, 1.76709656021757708569328754194, 2.33556550242256407561875745858, 2.72483366108877983658896412610, 2.89651627429618269030921966383, 3.62277151803108532434416368869, 3.73784388367155685194256090339, 3.83831486766162639447400563118, 3.85790963983727758183321413905, 4.42465799549446673700899685517, 4.56382031935202535692821598665, 4.84961506982248912812360369223, 4.96119971002619649983511471878, 5.17743806527440331177963645611, 5.71533954470351759201949442004, 6.08640589843678129026770434084, 6.26985799216267851454384098850, 6.72354865415525116019950534868, 6.77545329552074397306886529855, 6.82912986996465439482740748315, 7.55340183112394784194766656539, 7.67607952565961795904728471905, 7.87641497334521749919895850929, 8.010039276281686126806313404292
