| L(s) = 1 | − 8·5-s + 20·7-s + 20·11-s − 20·17-s − 38·19-s − 40·23-s − 2·25-s − 160·35-s + 20·43-s − 160·47-s + 202·49-s − 160·55-s − 20·61-s − 20·73-s + 400·77-s + 140·83-s + 160·85-s + 304·95-s + 320·115-s − 400·119-s + 58·121-s + 344·125-s + 127-s + 131-s − 760·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 20/7·7-s + 1.81·11-s − 1.17·17-s − 2·19-s − 1.73·23-s − 0.0799·25-s − 4.57·35-s + 0.465·43-s − 3.40·47-s + 4.12·49-s − 2.90·55-s − 0.327·61-s − 0.273·73-s + 5.19·77-s + 1.68·83-s + 1.88·85-s + 16/5·95-s + 2.78·115-s − 3.36·119-s + 0.479·121-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s − 5.71·133-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1392734103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1392734103\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 250 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 482 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1622 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2162 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3890 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5762 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3170 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 718 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12182 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5042 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13010 T^{2} + p^{4} T^{4} \) |
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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
−8.939740459607827378890608954137, −8.275894058089597159868054884042, −8.182192272491092724713868251433, −7.71532754222860423244608047131, −7.70419885108029885613329107263, −6.88093828368026515409070534189, −6.65767758222565868448497478835, −6.21384291701002508986663312584, −5.85803290473776230659147779265, −5.09827804762983624473892201321, −4.71596957154463816108555574343, −4.42214055269399308662618963456, −4.23558673262339735914459964559, −3.71485947343092483477021546398, −3.61096462915241623382695676982, −2.34942641025673812072461627161, −2.11347530713007540101659571774, −1.54385997192990941027552130606, −1.29889896153095985114723183451, −0.085063885152692211467309290560,
0.085063885152692211467309290560, 1.29889896153095985114723183451, 1.54385997192990941027552130606, 2.11347530713007540101659571774, 2.34942641025673812072461627161, 3.61096462915241623382695676982, 3.71485947343092483477021546398, 4.23558673262339735914459964559, 4.42214055269399308662618963456, 4.71596957154463816108555574343, 5.09827804762983624473892201321, 5.85803290473776230659147779265, 6.21384291701002508986663312584, 6.65767758222565868448497478835, 6.88093828368026515409070534189, 7.70419885108029885613329107263, 7.71532754222860423244608047131, 8.182192272491092724713868251433, 8.275894058089597159868054884042, 8.939740459607827378890608954137
