| L(s) = 1 | + 3-s − 10·5-s − 57·7-s − 9·9-s + 10·11-s − 13·13-s − 10·15-s − 51·17-s − 38·19-s − 57·21-s + 155·23-s + 2·25-s + 8·27-s + 79·29-s + 16·31-s + 10·33-s + 570·35-s − 380·37-s − 13·39-s − 790·41-s + 296·43-s + 90·45-s + 200·47-s + 1.79e3·49-s − 51·51-s − 397·53-s − 100·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s − 0.894·5-s − 3.07·7-s − 1/3·9-s + 0.274·11-s − 0.277·13-s − 0.172·15-s − 0.727·17-s − 0.458·19-s − 0.592·21-s + 1.40·23-s + 0.0159·25-s + 0.0570·27-s + 0.505·29-s + 0.0926·31-s + 0.0527·33-s + 2.75·35-s − 1.68·37-s − 0.0533·39-s − 3.00·41-s + 1.04·43-s + 0.298·45-s + 0.620·47-s + 5.23·49-s − 0.140·51-s − 1.02·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 10 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 98 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 57 T + 1454 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + p T + 2268 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 155 T + 994 p T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 79 T + 13124 T^{2} - 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 16 T + 48318 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 380 T + 126078 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 296 T + 78966 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 200 T + 146846 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 397 T + 333572 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 680 T + 483894 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 939 T + 740138 T^{2} + 939 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 406 T + 735614 T^{2} + 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 106 T + 840030 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2226 T + 2380750 T^{2} - 2226 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.115591743301416199302857930347, −8.897181636962605483390146839630, −8.449458603357767068196691145450, −8.027192870226431855214496931993, −7.23296308135166431020012858388, −6.98916443232887175328796850054, −6.74210356246334583246455982095, −6.43675559353954060121423570727, −5.91827891666826328459040700990, −5.37967031715861439540260419790, −4.83885141795867023543641392160, −4.23292679634972068933911494695, −3.63389331023466556412701610115, −3.48102484450201908960288781950, −2.91386327102178614091579413267, −2.67356849369048313064078196400, −1.76574158876416607840065488590, −0.75228481363562075767106516236, 0, 0,
0.75228481363562075767106516236, 1.76574158876416607840065488590, 2.67356849369048313064078196400, 2.91386327102178614091579413267, 3.48102484450201908960288781950, 3.63389331023466556412701610115, 4.23292679634972068933911494695, 4.83885141795867023543641392160, 5.37967031715861439540260419790, 5.91827891666826328459040700990, 6.43675559353954060121423570727, 6.74210356246334583246455982095, 6.98916443232887175328796850054, 7.23296308135166431020012858388, 8.027192870226431855214496931993, 8.449458603357767068196691145450, 8.897181636962605483390146839630, 9.115591743301416199302857930347
