| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 11·5-s + 24·6-s + 9·7-s + 32·8-s + 27·9-s + 44·10-s − 11-s + 72·12-s + 16·13-s + 36·14-s + 66·15-s + 80·16-s + 9·17-s + 108·18-s − 38·19-s + 132·20-s + 54·21-s − 4·22-s − 4·23-s + 192·24-s − 91·25-s + 64·26-s + 108·27-s + 108·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.983·5-s + 1.63·6-s + 0.485·7-s + 1.41·8-s + 9-s + 1.39·10-s − 0.0274·11-s + 1.73·12-s + 0.341·13-s + 0.687·14-s + 1.13·15-s + 5/4·16-s + 0.128·17-s + 1.41·18-s − 0.458·19-s + 1.47·20-s + 0.561·21-s − 0.0387·22-s − 0.0362·23-s + 1.63·24-s − 0.727·25-s + 0.482·26-s + 0.769·27-s + 0.728·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.516024946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.516024946\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 11 T + 212 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 9 T + 638 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 62 p T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 16 T + 3366 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 9232 T^{2} - 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 23246 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20 T + 44510 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 238 T + 27606 T^{2} + 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 262 T + 111642 T^{2} + 262 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 122 T + 21170 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 427 T + 154842 T^{2} + 427 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 125 T + 85358 T^{2} - 125 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 140 T + 263342 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 96 T + 199030 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 551 T + 528156 T^{2} + 551 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 912 T + 595430 T^{2} + 912 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 704 T + 560174 T^{2} - 704 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 321 T + 792260 T^{2} + 321 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 512 T + 313422 T^{2} + 512 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1188 T + 1014838 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1776 T + 2197390 T^{2} - 1776 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1772 T + 2605974 T^{2} - 1772 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.34136937295188028637489609917, −13.25413459789335343612334396201, −12.53516722728363726844226326482, −12.02957222247992701176675439653, −11.38789091621700196972900974707, −10.84970970007202918160346935563, −10.08325723091276971941049600757, −9.924401103377014813166094444507, −8.834775000354151852475427811805, −8.706309531797578486500713309740, −7.54300557650389144118423495347, −7.51235573100062936865795403740, −6.35773127989627423622566118361, −6.09706864262547976809493156396, −5.12757779808974982543547151743, −4.71370743600030435475591747607, −3.63476975345330792273079822652, −3.30132756438866760058980356511, −1.98966080664974956209597676846, −1.83367891753032477625765752903,
1.83367891753032477625765752903, 1.98966080664974956209597676846, 3.30132756438866760058980356511, 3.63476975345330792273079822652, 4.71370743600030435475591747607, 5.12757779808974982543547151743, 6.09706864262547976809493156396, 6.35773127989627423622566118361, 7.51235573100062936865795403740, 7.54300557650389144118423495347, 8.706309531797578486500713309740, 8.834775000354151852475427811805, 9.924401103377014813166094444507, 10.08325723091276971941049600757, 10.84970970007202918160346935563, 11.38789091621700196972900974707, 12.02957222247992701176675439653, 12.53516722728363726844226326482, 13.25413459789335343612334396201, 13.34136937295188028637489609917
