| L(s) = 1 | + 2-s + 3-s + 6·5-s + 6-s − 3·7-s − 8-s + 3·9-s + 6·10-s − 3·14-s + 6·15-s − 16-s + 3·17-s + 3·18-s − 6·19-s − 3·21-s − 6·23-s − 24-s + 17·25-s + 8·27-s + 6·30-s + 3·34-s − 18·35-s − 3·37-s − 6·38-s − 6·40-s − 3·42-s − 43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 2.68·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 9-s + 1.89·10-s − 0.801·14-s + 1.54·15-s − 1/4·16-s + 0.727·17-s + 0.707·18-s − 1.37·19-s − 0.654·21-s − 1.25·23-s − 0.204·24-s + 17/5·25-s + 1.53·27-s + 1.09·30-s + 0.514·34-s − 3.04·35-s − 0.493·37-s − 0.973·38-s − 0.948·40-s − 0.462·42-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.785569646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.785569646\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−12.26395818682726632116059549512, −11.19697073523930542194592637034, −10.53990999773492967339176471733, −10.25064951354585636236305276574, −9.844311197465793993829672255421, −9.632752111841649690562806604446, −9.192205794854160367539989416599, −8.668468844428794053999956023319, −8.120769876406963313617146210040, −7.33979937657994471212318056198, −6.52630714919313276134151311773, −6.42917376145958731951745367094, −6.05073368177609842850232491163, −5.34852880142234575628543882209, −4.99783594429938193343624104777, −4.04817055877821881038495805772, −3.60231708309518999372236362043, −2.59365953319811390258767325425, −2.29470585204494744961248711066, −1.46212745385382789925937279808,
1.46212745385382789925937279808, 2.29470585204494744961248711066, 2.59365953319811390258767325425, 3.60231708309518999372236362043, 4.04817055877821881038495805772, 4.99783594429938193343624104777, 5.34852880142234575628543882209, 6.05073368177609842850232491163, 6.42917376145958731951745367094, 6.52630714919313276134151311773, 7.33979937657994471212318056198, 8.120769876406963313617146210040, 8.668468844428794053999956023319, 9.192205794854160367539989416599, 9.632752111841649690562806604446, 9.844311197465793993829672255421, 10.25064951354585636236305276574, 10.53990999773492967339176471733, 11.19697073523930542194592637034, 12.26395818682726632116059549512
