| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s − 8·14-s − 4·16-s − 12·17-s − 8·23-s − 8·28-s + 20·31-s − 8·32-s − 24·34-s − 20·41-s − 16·46-s − 8·47-s − 2·49-s + 40·62-s − 8·64-s − 24·68-s + 8·71-s − 20·73-s − 28·79-s − 40·82-s − 28·89-s − 16·92-s − 16·94-s + 20·97-s − 4·98-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s − 2.13·14-s − 16-s − 2.91·17-s − 1.66·23-s − 1.51·28-s + 3.59·31-s − 1.41·32-s − 4.11·34-s − 3.12·41-s − 2.35·46-s − 1.16·47-s − 2/7·49-s + 5.08·62-s − 64-s − 2.91·68-s + 0.949·71-s − 2.34·73-s − 3.15·79-s − 4.41·82-s − 2.96·89-s − 1.66·92-s − 1.65·94-s + 2.03·97-s − 0.404·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4462112195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4462112195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.671632871743074861267639454371, −8.840673059193597144670569822022, −8.831032880567445837712472912468, −8.249431749506320013405460716063, −8.085107724007899737535770948726, −7.09353897325785887548341595208, −6.75634869648055366208379328080, −6.63203457664901773289278052594, −6.33035114290747587140794483536, −5.90587953654790420190448298493, −5.49135307478256597520897035521, −4.62510904831370408658515021086, −4.50311049206574359300872126957, −4.36661479518166264265115554616, −3.58007478127175611038371481458, −3.05537090306069614944728035551, −2.90187725485257266717297813458, −2.19483216598064543239036093092, −1.69654807277499282291602406887, −0.17848268403632496330964072075,
0.17848268403632496330964072075, 1.69654807277499282291602406887, 2.19483216598064543239036093092, 2.90187725485257266717297813458, 3.05537090306069614944728035551, 3.58007478127175611038371481458, 4.36661479518166264265115554616, 4.50311049206574359300872126957, 4.62510904831370408658515021086, 5.49135307478256597520897035521, 5.90587953654790420190448298493, 6.33035114290747587140794483536, 6.63203457664901773289278052594, 6.75634869648055366208379328080, 7.09353897325785887548341595208, 8.085107724007899737535770948726, 8.249431749506320013405460716063, 8.831032880567445837712472912468, 8.840673059193597144670569822022, 9.671632871743074861267639454371
