| L(s) = 1 | − 2·3-s − 4·5-s + 44·7-s − 51·9-s + 40·11-s − 20·13-s + 8·15-s + 46·17-s + 38·19-s − 88·21-s − 220·23-s + 134·25-s + 158·27-s − 84·29-s − 84·31-s − 80·33-s − 176·35-s + 304·37-s + 40·39-s + 168·41-s − 372·43-s + 204·45-s + 584·47-s + 859·49-s − 92·51-s + 484·53-s − 160·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.357·5-s + 2.37·7-s − 1.88·9-s + 1.09·11-s − 0.426·13-s + 0.137·15-s + 0.656·17-s + 0.458·19-s − 0.914·21-s − 1.99·23-s + 1.07·25-s + 1.12·27-s − 0.537·29-s − 0.486·31-s − 0.422·33-s − 0.849·35-s + 1.35·37-s + 0.164·39-s + 0.639·41-s − 1.31·43-s + 0.675·45-s + 1.81·47-s + 2.50·49-s − 0.252·51-s + 1.25·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.886460734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.886460734\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 5 | $D_{4}$ | \( 1 + 4 T - 118 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 44 T + 1077 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 40 T + 2690 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 20 T + 3657 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 46 T + 259 p T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 220 T + 1483 p T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 16969 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 84 T + 31214 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 304 T + 121062 T^{2} - 304 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 168 T + 107698 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 372 T + 193238 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 584 T + 239342 T^{2} - 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 484 T + 255041 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1074 T + 697639 T^{2} + 1074 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 88 T + 437670 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1430 T + 1039839 T^{2} - 1430 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1848 T + 1569226 T^{2} - 1848 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1294 T + 1190691 T^{2} + 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 832 T + 593322 T^{2} - 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 528 T + 459970 T^{2} - 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1844 T + 1750754 T^{2} - 1844 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 364 T + 1606998 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.69583395761486561324246989521, −10.12918956229678069472855003212, −9.455245433095059482272996779012, −9.087272698770297342387532894434, −8.623087338056436291539441504435, −8.197042909626086281231884747493, −7.84110548998567692004543014971, −7.65505507856315253875470226391, −6.93360859296895455037897176502, −6.18343718823859911350267727689, −5.92173372085479656496590720817, −5.37627298714958766071672380536, −4.89457850234578566118572786117, −4.62146248313441033775948832393, −3.76532442262046395874959211379, −3.45444224633072213648202585261, −2.33025033330693475963870978157, −2.11515222307401251129031749914, −1.11760449648028744145254785296, −0.58569813048790144854262308024,
0.58569813048790144854262308024, 1.11760449648028744145254785296, 2.11515222307401251129031749914, 2.33025033330693475963870978157, 3.45444224633072213648202585261, 3.76532442262046395874959211379, 4.62146248313441033775948832393, 4.89457850234578566118572786117, 5.37627298714958766071672380536, 5.92173372085479656496590720817, 6.18343718823859911350267727689, 6.93360859296895455037897176502, 7.65505507856315253875470226391, 7.84110548998567692004543014971, 8.197042909626086281231884747493, 8.623087338056436291539441504435, 9.087272698770297342387532894434, 9.455245433095059482272996779012, 10.12918956229678069472855003212, 10.69583395761486561324246989521
