| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s − 8·14-s − 4·16-s + 4·17-s − 8·23-s + 6·25-s − 8·28-s + 4·31-s − 8·32-s + 8·34-s − 4·41-s − 16·46-s + 24·47-s − 2·49-s + 12·50-s + 8·62-s − 8·64-s + 8·68-s − 24·71-s − 12·73-s + 20·79-s − 8·82-s + 20·89-s − 16·92-s + 48·94-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s − 2.13·14-s − 16-s + 0.970·17-s − 1.66·23-s + 6/5·25-s − 1.51·28-s + 0.718·31-s − 1.41·32-s + 1.37·34-s − 0.624·41-s − 2.35·46-s + 3.50·47-s − 2/7·49-s + 1.69·50-s + 1.01·62-s − 64-s + 0.970·68-s − 2.84·71-s − 1.40·73-s + 2.25·79-s − 0.883·82-s + 2.11·89-s − 1.66·92-s + 4.95·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.408882848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.408882848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−14.82928336017101520461775056403, −14.27967789830314087067313413679, −13.63584948772805157440499549654, −13.50751769492664210204676631508, −12.70063404200723784944113138791, −12.27510916837260584493795738950, −12.08350832093766156717979241911, −11.29820456022791443126402878567, −10.24089300325458465964867268631, −10.22129357345684336042357840687, −9.232706728984305294883789634900, −8.780433665194772995003156464804, −7.73033600871715252944656625446, −7.04757015115889787904429590464, −6.19224423072169904665005262741, −5.99689698164512799731262664257, −5.05733945414218989480480092363, −4.13547310437713837033869561692, −3.42160512196739537031611417666, −2.65049428307336927197666154875,
2.65049428307336927197666154875, 3.42160512196739537031611417666, 4.13547310437713837033869561692, 5.05733945414218989480480092363, 5.99689698164512799731262664257, 6.19224423072169904665005262741, 7.04757015115889787904429590464, 7.73033600871715252944656625446, 8.780433665194772995003156464804, 9.232706728984305294883789634900, 10.22129357345684336042357840687, 10.24089300325458465964867268631, 11.29820456022791443126402878567, 12.08350832093766156717979241911, 12.27510916837260584493795738950, 12.70063404200723784944113138791, 13.50751769492664210204676631508, 13.63584948772805157440499549654, 14.27967789830314087067313413679, 14.82928336017101520461775056403
