| L(s) = 1 | + 2·2-s + 16·4-s − 18·5-s + 77·7-s + 88·8-s − 36·10-s + 194·11-s + 154·14-s + 176·16-s + 420·17-s + 453·19-s − 288·20-s + 388·22-s − 112·23-s − 409·25-s + 1.23e3·28-s − 2.08e3·29-s − 2.01e3·31-s + 1.40e3·32-s + 840·34-s − 1.38e3·35-s + 1.07e3·37-s + 906·38-s − 1.58e3·40-s − 2.17e3·43-s + 3.10e3·44-s − 224·46-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 4-s − 0.719·5-s + 11/7·7-s + 11/8·8-s − 0.359·10-s + 1.60·11-s + 0.785·14-s + 0.687·16-s + 1.45·17-s + 1.25·19-s − 0.719·20-s + 0.801·22-s − 0.211·23-s − 0.654·25-s + 11/7·28-s − 2.47·29-s − 2.10·31-s + 11/8·32-s + 0.726·34-s − 1.13·35-s + 0.785·37-s + 0.627·38-s − 0.989·40-s − 1.17·43-s + 1.60·44-s − 0.105·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(4.394453146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.394453146\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 11 p T + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T - 3 p^{2} T^{2} - p^{5} T^{3} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 18 T + 733 T^{2} + 18 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T + 22995 T^{2} - 194 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 30047 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 420 T + 142321 T^{2} - 420 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 453 T + 198724 T^{2} - 453 p^{4} T^{3} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 112 T - 267297 T^{2} + 112 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1040 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2019 T + 2282308 T^{2} + 2019 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1075 T - 718536 T^{2} - 1075 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3945974 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 1087 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3750 T + 9567181 T^{2} + 3750 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2200 T - 3050481 T^{2} + 2200 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9264 T + 40724593 T^{2} - 9264 p^{4} T^{3} + p^{8} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 1212 T + 14335489 T^{2} - 1212 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2375 T - 14510496 T^{2} + 2375 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8938 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 15807 T + 111685324 T^{2} - 15807 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8147 T + 27423528 T^{2} + 8147 p^{4} T^{3} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50356694 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 23628 T + 248836369 T^{2} + 23628 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 164817362 T^{2} + p^{8} T^{4} \) |
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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
−14.32089875219252990325660925862, −14.30205028223117832967668188357, −13.50585102123787419582654485092, −12.70519762275165379014290690698, −12.11298026519289589950522316057, −11.44868150622647619493555261578, −11.21602206492184614854761548614, −11.14258617817962031250964548389, −9.723511403929811557034654453515, −9.559058614213774058229265385712, −8.167298449682026547824708941306, −7.961634943721714260192242952539, −7.25874833610880380314019416114, −6.80610778064116637107417731547, −5.51182928415277101191683389274, −5.20314021487325835480171690536, −3.89739780964248356559085424668, −3.71128534142391342004489497675, −1.90543105528283403944291115588, −1.32929890044565794409137569448,
1.32929890044565794409137569448, 1.90543105528283403944291115588, 3.71128534142391342004489497675, 3.89739780964248356559085424668, 5.20314021487325835480171690536, 5.51182928415277101191683389274, 6.80610778064116637107417731547, 7.25874833610880380314019416114, 7.961634943721714260192242952539, 8.167298449682026547824708941306, 9.559058614213774058229265385712, 9.723511403929811557034654453515, 11.14258617817962031250964548389, 11.21602206492184614854761548614, 11.44868150622647619493555261578, 12.11298026519289589950522316057, 12.70519762275165379014290690698, 13.50585102123787419582654485092, 14.30205028223117832967668188357, 14.32089875219252990325660925862
