| L(s) = 1 | − 2·3-s − 2·7-s − 7·9-s + 4·21-s + 14·23-s + 20·27-s − 18·29-s + 10·41-s + 18·43-s + 18·47-s − 23·49-s + 18·61-s + 14·63-s + 24·67-s − 28·69-s + 20·81-s + 10·83-s + 36·87-s + 26·89-s − 34·101-s + 32·103-s + 54·107-s − 6·109-s + 8·121-s − 20·123-s + 127-s − 36·129-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s − 7/3·9-s + 0.872·21-s + 2.91·23-s + 3.84·27-s − 3.34·29-s + 1.56·41-s + 2.74·43-s + 2.62·47-s − 3.28·49-s + 2.30·61-s + 1.76·63-s + 2.93·67-s − 3.37·69-s + 20/9·81-s + 1.09·83-s + 3.85·87-s + 2.75·89-s − 3.38·101-s + 3.15·103-s + 5.22·107-s − 0.574·109-s + 8/11·121-s − 1.80·123-s + 0.0887·127-s − 3.16·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680715788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680715788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 8 T^{2} + 238 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 318 T^{4} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 78 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 238 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 3198 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 60 T^{2} + 2358 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 5 T + 77 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 9 T + 83 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 4 T^{2} + 5302 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 6606 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 9 T + 131 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 216 T^{2} + 21246 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 32 T^{2} + 10414 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 228 T^{2} + 24198 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 5 T + 71 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 13 T + 209 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 22414 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.03754352647157514433220241587, −5.79249713901146991253597860079, −5.57932400918253796540321020638, −5.36833677564909285251493962628, −5.28881237529974181631551763333, −4.98473523052769131967442592354, −4.94800453690826299877052876477, −4.85187350533789480600197359159, −4.39183998420279698967653721832, −4.14868038687912823687012102635, −3.82675483175360087059479578380, −3.60497298080258198980052572396, −3.60276573024960175612405484143, −3.36775829916124066994858965705, −3.06061574201469835412014948086, −2.78247817031101696442145548398, −2.64004263205035883534809672433, −2.40714561161291904105615398088, −2.17942471455077353328213801857, −2.01478706557906705398709420007, −1.54148104130514304866629289772, −0.915076873962790502991962371638, −0.829507448402520658462341994900, −0.58022852315035591905303209686, −0.33615804984519112089974013293,
0.33615804984519112089974013293, 0.58022852315035591905303209686, 0.829507448402520658462341994900, 0.915076873962790502991962371638, 1.54148104130514304866629289772, 2.01478706557906705398709420007, 2.17942471455077353328213801857, 2.40714561161291904105615398088, 2.64004263205035883534809672433, 2.78247817031101696442145548398, 3.06061574201469835412014948086, 3.36775829916124066994858965705, 3.60276573024960175612405484143, 3.60497298080258198980052572396, 3.82675483175360087059479578380, 4.14868038687912823687012102635, 4.39183998420279698967653721832, 4.85187350533789480600197359159, 4.94800453690826299877052876477, 4.98473523052769131967442592354, 5.28881237529974181631551763333, 5.36833677564909285251493962628, 5.57932400918253796540321020638, 5.79249713901146991253597860079, 6.03754352647157514433220241587
