| L(s) = 1 | − 4-s + 16-s − 6·17-s − 12·23-s + 25-s − 6·29-s + 20·43-s + 10·49-s − 6·53-s − 14·61-s − 64-s + 6·68-s − 8·79-s + 12·92-s − 100-s + 30·101-s − 28·103-s + 12·107-s + 6·113-s + 6·116-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/4·16-s − 1.45·17-s − 2.50·23-s + 1/5·25-s − 1.11·29-s + 3.04·43-s + 10/7·49-s − 0.824·53-s − 1.79·61-s − 1/8·64-s + 0.727·68-s − 0.900·79-s + 1.25·92-s − 0.0999·100-s + 2.98·101-s − 2.75·103-s + 1.16·107-s + 0.564·113-s + 0.557·116-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9253764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7293240144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7293240144\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.994131226139116208460147131087, −8.775347656742341972043894041688, −7.947305135300674233312556026942, −7.77396774518425991021370591890, −7.68638894058128807497550973040, −6.93129654183147120621234193936, −6.75969561148304226372221558972, −5.97086564337813197198563685615, −5.95286256421286728361577274680, −5.67199493111076621280458218394, −4.98881721024624112984027305687, −4.42244291550634203537230401192, −4.28109814095154246856433026863, −3.94817747707020312286286318964, −3.40574834170989541419037801669, −2.78703387214238670327121021428, −2.18425003754090017384578551295, −1.99678281305320550289416070561, −1.15507579099097851913742935470, −0.28375944327736613717004271617,
0.28375944327736613717004271617, 1.15507579099097851913742935470, 1.99678281305320550289416070561, 2.18425003754090017384578551295, 2.78703387214238670327121021428, 3.40574834170989541419037801669, 3.94817747707020312286286318964, 4.28109814095154246856433026863, 4.42244291550634203537230401192, 4.98881721024624112984027305687, 5.67199493111076621280458218394, 5.95286256421286728361577274680, 5.97086564337813197198563685615, 6.75969561148304226372221558972, 6.93129654183147120621234193936, 7.68638894058128807497550973040, 7.77396774518425991021370591890, 7.947305135300674233312556026942, 8.775347656742341972043894041688, 8.994131226139116208460147131087
