| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s − 2·11-s + 4·13-s + 4·14-s + 5·16-s + 4·19-s − 4·22-s + 2·25-s + 8·26-s + 6·28-s + 12·29-s + 4·31-s + 6·32-s + 4·37-s + 8·38-s − 8·43-s − 6·44-s − 12·47-s + 3·49-s + 4·50-s + 12·52-s − 12·53-s + 8·56-s + 24·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s + 0.917·19-s − 0.852·22-s + 2/5·25-s + 1.56·26-s + 1.13·28-s + 2.22·29-s + 0.718·31-s + 1.06·32-s + 0.657·37-s + 1.29·38-s − 1.21·43-s − 0.904·44-s − 1.75·47-s + 3/7·49-s + 0.565·50-s + 1.66·52-s − 1.64·53-s + 1.06·56-s + 3.15·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.662465043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.662465043\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.972633826112330311012347339239, −9.584280973472457077532098453311, −8.735042526922977639380233643399, −8.440896253551250356218315482221, −8.004935419709259562790123930571, −7.923337773819737726126142632621, −7.19714710749690088289298432029, −6.73412951282827252055176071913, −6.28669996459717867188376423926, −6.24470693102291441432326060408, −5.33866051183421485238341903525, −5.16136149441923229926405952281, −4.74963844284293610562660961487, −4.43692597497406014978953032748, −3.60788259867162602760732647188, −3.43660155055806753053385219239, −2.76072760305783002965781308600, −2.36039802789963555270986036165, −1.48466756456764650959540372903, −0.998111015528653354260886153503,
0.998111015528653354260886153503, 1.48466756456764650959540372903, 2.36039802789963555270986036165, 2.76072760305783002965781308600, 3.43660155055806753053385219239, 3.60788259867162602760732647188, 4.43692597497406014978953032748, 4.74963844284293610562660961487, 5.16136149441923229926405952281, 5.33866051183421485238341903525, 6.24470693102291441432326060408, 6.28669996459717867188376423926, 6.73412951282827252055176071913, 7.19714710749690088289298432029, 7.923337773819737726126142632621, 8.004935419709259562790123930571, 8.440896253551250356218315482221, 8.735042526922977639380233643399, 9.584280973472457077532098453311, 9.972633826112330311012347339239
