| L(s) = 1 | − 3·4-s + 2·5-s − 3·9-s − 8·11-s + 6·16-s + 6·19-s − 6·20-s + 25-s − 20·29-s + 28·31-s + 9·36-s + 24·44-s − 6·45-s + 10·49-s − 16·55-s − 28·59-s + 12·61-s − 10·64-s − 8·71-s − 18·76-s − 44·79-s + 12·80-s + 6·81-s − 8·89-s + 12·95-s + 24·99-s − 3·100-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 0.894·5-s − 9-s − 2.41·11-s + 3/2·16-s + 1.37·19-s − 1.34·20-s + 1/5·25-s − 3.71·29-s + 5.02·31-s + 3/2·36-s + 3.61·44-s − 0.894·45-s + 10/7·49-s − 2.15·55-s − 3.64·59-s + 1.53·61-s − 5/4·64-s − 0.949·71-s − 2.06·76-s − 4.95·79-s + 1.34·80-s + 2/3·81-s − 0.847·89-s + 1.23·95-s + 2.41·99-s − 0.299·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.133121287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133121287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{3} \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | \( ( 1 - T )^{6} \) |
| good | 7 | \( 1 - 10 T^{2} + 95 T^{4} - 780 T^{6} + 95 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 34 T^{2} + 359 T^{4} - 2172 T^{6} + 359 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 42 T^{2} + 815 T^{4} - 13196 T^{6} + 815 p^{2} T^{8} - 42 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( 1 - 94 T^{2} + 3999 T^{4} - 108772 T^{6} + 3999 p^{2} T^{8} - 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 10 T + 99 T^{2} + 540 T^{3} + 99 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 - 14 T + 121 T^{2} - 716 T^{3} + 121 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 50 T^{2} + 1207 T^{4} + 100 T^{6} + 1207 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 + 107 T^{2} - 16 T^{3} + 107 p T^{4} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 150 T^{2} + 9335 T^{4} - 407060 T^{6} + 9335 p^{2} T^{8} - 150 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 174 T^{2} + 16271 T^{4} - 934532 T^{6} + 16271 p^{2} T^{8} - 174 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 146 T^{2} + 10071 T^{4} - 521948 T^{6} + 10071 p^{2} T^{8} - 146 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 14 T + 229 T^{2} + 1692 T^{3} + 229 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 67 | \( 1 - 274 T^{2} + 36103 T^{4} - 2962972 T^{6} + 36103 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 4 T + 133 T^{2} + 504 T^{3} + 133 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 246 T^{2} + 32063 T^{4} - 2771828 T^{6} + 32063 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 22 T + 361 T^{2} + 3676 T^{3} + 361 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 258 T^{2} + 38055 T^{4} - 3905724 T^{6} + 38055 p^{2} T^{8} - 258 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 4 T + 219 T^{2} + 792 T^{3} + 219 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 186 T^{2} + 22479 T^{4} - 2639468 T^{6} + 22479 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.64752859210790084458353382440, −5.55335048129087029829607774477, −5.53948080791930392039548549711, −5.19134533922343087387048329086, −5.19031312895059595666947136059, −4.91783412681503540662999688118, −4.77654619858320528935112950538, −4.49345870289209515427100951387, −4.41686992822531684825909464108, −4.38545624950981318904524002984, −3.93165019501242617773803477466, −3.77470426840634419183457133338, −3.61982911947592196133729640510, −3.42759277013220754525114901421, −2.92548584134605572906950871301, −2.81103339339700962416586161667, −2.74688008579784076951100849490, −2.65442586363393421800977762663, −2.65088112691561533377986155054, −1.94944813339847646432386961427, −1.73209085751887850125699977122, −1.40706244727869401407820124805, −1.26519813328542062943859763215, −0.46886071688508549989339828443, −0.41362594095533337292843002956,
0.41362594095533337292843002956, 0.46886071688508549989339828443, 1.26519813328542062943859763215, 1.40706244727869401407820124805, 1.73209085751887850125699977122, 1.94944813339847646432386961427, 2.65088112691561533377986155054, 2.65442586363393421800977762663, 2.74688008579784076951100849490, 2.81103339339700962416586161667, 2.92548584134605572906950871301, 3.42759277013220754525114901421, 3.61982911947592196133729640510, 3.77470426840634419183457133338, 3.93165019501242617773803477466, 4.38545624950981318904524002984, 4.41686992822531684825909464108, 4.49345870289209515427100951387, 4.77654619858320528935112950538, 4.91783412681503540662999688118, 5.19031312895059595666947136059, 5.19134533922343087387048329086, 5.53948080791930392039548549711, 5.55335048129087029829607774477, 5.64752859210790084458353382440
Plot not available for L-functions of degree greater than 10.
