| L(s) = 1 | − 2·2-s + 4-s − 24·5-s − 12·8-s + 48·10-s + 44·11-s − 26·13-s − 11·16-s − 164·17-s + 48·19-s − 24·20-s − 88·22-s − 8·23-s + 238·25-s + 52·26-s − 404·29-s + 40·31-s + 170·32-s + 328·34-s − 100·37-s − 96·38-s + 288·40-s − 200·41-s − 616·43-s + 44·44-s + 16·46-s + 324·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/8·4-s − 2.14·5-s − 0.530·8-s + 1.51·10-s + 1.20·11-s − 0.554·13-s − 0.171·16-s − 2.33·17-s + 0.579·19-s − 0.268·20-s − 0.852·22-s − 0.0725·23-s + 1.90·25-s + 0.392·26-s − 2.58·29-s + 0.231·31-s + 0.939·32-s + 1.65·34-s − 0.444·37-s − 0.409·38-s + 1.13·40-s − 0.761·41-s − 2.18·43-s + 0.150·44-s + 0.0512·46-s + 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 24 T + 338 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 90 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 p T + 1130 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 16326 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 48 T + 14238 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T - 7906 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 404 T + 81518 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 40 T + 50518 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 100 T + 92830 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 200 T + 24138 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 616 T + 216022 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 324 T + 219554 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 102878 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 140 T + 393258 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 628 T + 293614 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 472 T + 252622 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 428 T + 662834 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 900 T + 899670 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 924318 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1388 T + 1567866 T^{2} - 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 960 T + 1134938 T^{2} + 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 532 T + 1218502 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.63377013402291728405151092866, −11.93035959339855951985913242629, −11.63886287583979460261011077800, −11.42552329360681036081902905990, −10.95812536459132635329218249832, −10.08319105319050334074844463645, −9.294077543147806545208936945520, −9.108916971155018778334359686492, −8.433134935920717875958997297661, −8.063073205285482351626722771698, −7.21056578691358615986836758512, −6.97106369438775344828064583840, −6.32570344893574209253582398687, −5.17572821030091735397467274800, −4.30386547431927186983213540636, −3.95082809222005994297292905025, −3.13310505933253413178456107533, −1.82422593312439725647634377140, 0, 0,
1.82422593312439725647634377140, 3.13310505933253413178456107533, 3.95082809222005994297292905025, 4.30386547431927186983213540636, 5.17572821030091735397467274800, 6.32570344893574209253582398687, 6.97106369438775344828064583840, 7.21056578691358615986836758512, 8.063073205285482351626722771698, 8.433134935920717875958997297661, 9.108916971155018778334359686492, 9.294077543147806545208936945520, 10.08319105319050334074844463645, 10.95812536459132635329218249832, 11.42552329360681036081902905990, 11.63886287583979460261011077800, 11.93035959339855951985913242629, 12.63377013402291728405151092866
