| L(s) = 1 | + 6·2-s + 16·4-s + 39·7-s + 24·8-s + 39·11-s − 26·13-s + 234·14-s + 48·16-s + 27·17-s − 153·19-s + 234·22-s + 57·23-s + 58·25-s − 156·26-s + 624·28-s − 69·29-s + 192·32-s + 162·34-s − 69·37-s − 918·38-s + 681·41-s + 85·43-s + 624·44-s + 342·46-s + 671·49-s + 348·50-s − 416·52-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 2.10·7-s + 1.06·8-s + 1.06·11-s − 0.554·13-s + 4.46·14-s + 3/4·16-s + 0.385·17-s − 1.84·19-s + 2.26·22-s + 0.516·23-s + 0.463·25-s − 1.17·26-s + 4.21·28-s − 0.441·29-s + 1.06·32-s + 0.817·34-s − 0.306·37-s − 3.91·38-s + 2.59·41-s + 0.301·43-s + 2.13·44-s + 1.09·46-s + 1.95·49-s + 0.984·50-s − 1.10·52-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
| \(L(2)\) |
\(\approx\) |
\(8.223658568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.223658568\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T + 5 p^{2} T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 58 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 39 T + 850 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 39 T + 1838 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 27 T - 4184 T^{2} - 27 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 153 T + 14662 T^{2} + 153 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 57 T - 8918 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 69 T - 19628 T^{2} + 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54290 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 69 T + 52240 T^{2} + 69 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 681 T + 223508 T^{2} - 681 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 85 T - 72282 T^{2} - 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90034 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 426 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 33 T + 205742 T^{2} - 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 17 T - 226692 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 285 T + 327838 T^{2} - 285 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 1011 T + 698618 T^{2} + 1011 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 231166 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1244 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 962026 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 531 T + 798956 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2139 T + 2437780 T^{2} - 2139 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.09154426700337427850390128865, −13.01221302977528308469550986403, −12.42367806003130677447226061151, −11.98625923597066009433965016714, −11.22647456739344155084107078590, −11.21467285262399969721925163157, −10.50909562866011766445453706213, −9.710015530470312956424307225656, −8.765046303144688749093042001288, −8.542697695022497590944451970739, −7.54048570399965689406601754084, −7.31787021205226107606993654545, −6.01771565550680087053531216290, −6.00985419631430106998444668079, −4.86016382212429959101308102599, −4.64161844728261179857798511611, −4.26309525114901177612276458403, −3.37837983410957824319240141260, −2.25995207767357435164367621395, −1.35351404865054548894972742424,
1.35351404865054548894972742424, 2.25995207767357435164367621395, 3.37837983410957824319240141260, 4.26309525114901177612276458403, 4.64161844728261179857798511611, 4.86016382212429959101308102599, 6.00985419631430106998444668079, 6.01771565550680087053531216290, 7.31787021205226107606993654545, 7.54048570399965689406601754084, 8.542697695022497590944451970739, 8.765046303144688749093042001288, 9.710015530470312956424307225656, 10.50909562866011766445453706213, 11.21467285262399969721925163157, 11.22647456739344155084107078590, 11.98625923597066009433965016714, 12.42367806003130677447226061151, 13.01221302977528308469550986403, 13.09154426700337427850390128865
