| L(s) = 1 | + 4·2-s + 10·4-s + 4·7-s + 20·8-s + 8·11-s + 16·14-s + 35·16-s + 32·22-s + 4·23-s − 2·25-s + 40·28-s + 8·29-s + 4·31-s + 56·32-s + 4·37-s − 8·41-s + 8·43-s + 80·44-s + 16·46-s − 12·47-s + 10·49-s − 8·50-s + 12·53-s + 80·56-s + 32·58-s − 16·59-s + 16·62-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 1.51·7-s + 7.07·8-s + 2.41·11-s + 4.27·14-s + 35/4·16-s + 6.82·22-s + 0.834·23-s − 2/5·25-s + 7.55·28-s + 1.48·29-s + 0.718·31-s + 9.89·32-s + 0.657·37-s − 1.24·41-s + 1.21·43-s + 12.0·44-s + 2.35·46-s − 1.75·47-s + 10/7·49-s − 1.13·50-s + 1.64·53-s + 10.6·56-s + 4.20·58-s − 2.08·59-s + 2.03·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(95.57712024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(95.57712024\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 2 T^{2} + 8 T^{3} - 6 T^{4} + 8 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 8 T + 50 T^{2} - 216 T^{3} + 810 T^{4} - 216 p T^{5} + 50 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 22 T^{2} - 56 T^{3} + 210 T^{4} - 56 p T^{5} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 38 T^{2} + 56 T^{3} + 690 T^{4} + 56 p T^{5} + 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 28 T^{2} + 128 T^{3} + 306 T^{4} + 128 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 4 T + 58 T^{2} - 268 T^{3} + 1666 T^{4} - 268 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 70 T^{2} - 508 T^{3} + 2506 T^{4} - 508 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 58 T^{2} + 264 T^{3} - 1878 T^{4} + 264 p T^{5} + 58 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 170 T^{2} + 1332 T^{3} + 10914 T^{4} + 1332 p T^{5} + 170 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 12 T + 182 T^{2} - 1284 T^{3} + 12234 T^{4} - 1284 p T^{5} + 182 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 16 T + 260 T^{2} + 2448 T^{3} + 22950 T^{4} + 2448 p T^{5} + 260 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 82 T^{2} - 216 T^{3} + 4506 T^{4} - 216 p T^{5} + 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 24 T + 202 T^{2} - 232 T^{3} - 5814 T^{4} - 232 p T^{5} + 202 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 4 T + 176 T^{2} + 404 T^{3} + 15326 T^{4} + 404 p T^{5} + 176 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 12 T + 256 T^{2} - 2084 T^{3} + 26910 T^{4} - 2084 p T^{5} + 256 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 28 T + 520 T^{2} - 84 p T^{3} + 68142 T^{4} - 84 p^{2} T^{5} + 520 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 8 T + 44 T^{2} + 8 T^{3} + 3890 T^{4} + 8 p T^{5} + 44 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 8 T + 110 T^{2} + 1248 T^{3} - 7566 T^{4} + 1248 p T^{5} + 110 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 8 T + 118 T^{2} + 400 T^{3} + 4946 T^{4} + 400 p T^{5} + 118 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.31274070638982889649220669698, −5.80424936515362052674980389513, −5.70995432144792721792390857613, −5.69534226912036936260464614450, −5.27886562849135162874018862818, −4.95503123074504498599563477888, −4.91185568874752297320930377932, −4.86149267969707143959088249959, −4.66754054301739742059680617517, −4.23601720073400727847612449541, −4.20930098823176892855867515336, −3.98073634864565683740368948395, −3.89027398506576580254816825098, −3.43722865987521530131451610785, −3.31551483838576175380669412449, −3.18554290986274477879877988539, −3.01071144213378302324550733193, −2.37452031293872583595522418169, −2.28931047164323532437714401555, −2.07281686113120558942125832060, −2.02451160156223353432499671067, −1.37350238077133991692520456472, −1.26839990713892651594618424083, −0.986642704637210167038907310081, −0.73424577952363051266509882954,
0.73424577952363051266509882954, 0.986642704637210167038907310081, 1.26839990713892651594618424083, 1.37350238077133991692520456472, 2.02451160156223353432499671067, 2.07281686113120558942125832060, 2.28931047164323532437714401555, 2.37452031293872583595522418169, 3.01071144213378302324550733193, 3.18554290986274477879877988539, 3.31551483838576175380669412449, 3.43722865987521530131451610785, 3.89027398506576580254816825098, 3.98073634864565683740368948395, 4.20930098823176892855867515336, 4.23601720073400727847612449541, 4.66754054301739742059680617517, 4.86149267969707143959088249959, 4.91185568874752297320930377932, 4.95503123074504498599563477888, 5.27886562849135162874018862818, 5.69534226912036936260464614450, 5.70995432144792721792390857613, 5.80424936515362052674980389513, 6.31274070638982889649220669698
