| L(s) = 1 | + 4·3-s + 4·7-s + 8·9-s + 2·13-s + 10·17-s − 8·19-s + 16·21-s + 4·23-s + 12·27-s − 2·37-s + 8·39-s − 12·43-s − 4·47-s + 8·49-s + 40·51-s + 14·53-s − 32·57-s − 8·59-s − 8·61-s + 32·63-s − 20·67-s + 16·69-s + 6·73-s − 32·79-s + 23·81-s − 4·83-s + 8·91-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.51·7-s + 8/3·9-s + 0.554·13-s + 2.42·17-s − 1.83·19-s + 3.49·21-s + 0.834·23-s + 2.30·27-s − 0.328·37-s + 1.28·39-s − 1.82·43-s − 0.583·47-s + 8/7·49-s + 5.60·51-s + 1.92·53-s − 4.23·57-s − 1.04·59-s − 1.02·61-s + 4.03·63-s − 2.44·67-s + 1.92·69-s + 0.702·73-s − 3.60·79-s + 23/9·81-s − 0.439·83-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.119774296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.119774296\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.43811160386740641218102026062, −9.893377643754679860474305996525, −9.626181625712484079981184380971, −8.920870497237094801669080271307, −8.557331739005640020914052836296, −8.451078891250959854399863320179, −8.189337562407795786323827596724, −7.64253639911142376629320775619, −7.25470210770353773462164029968, −6.92319142458831143781788330724, −5.85994756202793568258382430565, −5.81953384772325011291471062080, −4.71431384262467874665852686988, −4.70361057410788386802092093070, −3.93607327561960163287629937021, −3.24478951700895614579558686438, −3.16453854697710220546237269357, −2.37495752714461338768708010697, −1.68588669912474800784623989980, −1.32326887397239178546294212757,
1.32326887397239178546294212757, 1.68588669912474800784623989980, 2.37495752714461338768708010697, 3.16453854697710220546237269357, 3.24478951700895614579558686438, 3.93607327561960163287629937021, 4.70361057410788386802092093070, 4.71431384262467874665852686988, 5.81953384772325011291471062080, 5.85994756202793568258382430565, 6.92319142458831143781788330724, 7.25470210770353773462164029968, 7.64253639911142376629320775619, 8.189337562407795786323827596724, 8.451078891250959854399863320179, 8.557331739005640020914052836296, 8.920870497237094801669080271307, 9.626181625712484079981184380971, 9.893377643754679860474305996525, 10.43811160386740641218102026062
