| L(s) = 1 | + 2·5-s − 10·7-s + 10·11-s + 50·17-s − 38·19-s − 20·23-s − 47·25-s − 20·35-s − 10·43-s + 10·47-s − 23·49-s + 20·55-s + 190·61-s − 50·73-s − 100·77-s − 260·83-s + 100·85-s − 76·95-s − 40·115-s − 500·119-s − 167·121-s − 146·125-s + 127-s + 131-s + 380·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2/5·5-s − 1.42·7-s + 0.909·11-s + 2.94·17-s − 2·19-s − 0.869·23-s − 1.87·25-s − 4/7·35-s − 0.232·43-s + 0.212·47-s − 0.469·49-s + 4/11·55-s + 3.11·61-s − 0.684·73-s − 1.29·77-s − 3.13·83-s + 1.17·85-s − 4/5·95-s − 0.347·115-s − 4.20·119-s − 1.38·121-s − 1.16·125-s + 0.00787·127-s + 0.00763·131-s + 20/7·133-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07848296464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07848296464\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 25 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 118 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 122 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2090 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1562 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 4970 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 95 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3190 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10682 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 358 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18530 T^{2} + p^{4} T^{4} \) |
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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
−9.053497902260648341853819117706, −8.372005621612251539548339986581, −8.139385125061486550321219101794, −7.67887131449059308028296017835, −7.41801470267789191337744283757, −6.67675737776153380221074464151, −6.60701173728588852828043724310, −6.07470676648128397419831805871, −5.96178034572147365806093364711, −5.34850259464781383766638675915, −5.23806762985360126145056278459, −4.15836216879720179864767444652, −4.10134878980556283757770958286, −3.56809313776093622397886844053, −3.39292076129647853993475819845, −2.53780423572891821809319949412, −2.35198206552453688876835269479, −1.41182019977938210529965669826, −1.25728752768970767778708431025, −0.06544891805562808766469160398,
0.06544891805562808766469160398, 1.25728752768970767778708431025, 1.41182019977938210529965669826, 2.35198206552453688876835269479, 2.53780423572891821809319949412, 3.39292076129647853993475819845, 3.56809313776093622397886844053, 4.10134878980556283757770958286, 4.15836216879720179864767444652, 5.23806762985360126145056278459, 5.34850259464781383766638675915, 5.96178034572147365806093364711, 6.07470676648128397419831805871, 6.60701173728588852828043724310, 6.67675737776153380221074464151, 7.41801470267789191337744283757, 7.67887131449059308028296017835, 8.139385125061486550321219101794, 8.372005621612251539548339986581, 9.053497902260648341853819117706
