| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 4·7-s + 4·8-s − 2·10-s + 6·11-s − 6·13-s − 8·14-s + 8·16-s + 6·17-s − 7·19-s − 2·20-s + 12·22-s − 8·23-s − 12·26-s − 8·28-s + 7·29-s + 18·31-s + 8·32-s + 12·34-s + 4·35-s − 4·37-s − 14·38-s − 4·40-s + 6·41-s − 10·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 1.51·7-s + 1.41·8-s − 0.632·10-s + 1.80·11-s − 1.66·13-s − 2.13·14-s + 2·16-s + 1.45·17-s − 1.60·19-s − 0.447·20-s + 2.55·22-s − 1.66·23-s − 2.35·26-s − 1.51·28-s + 1.29·29-s + 3.23·31-s + 1.41·32-s + 2.05·34-s + 0.676·35-s − 0.657·37-s − 2.27·38-s − 0.632·40-s + 0.937·41-s − 1.52·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.670598304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.670598304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 7 T - 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 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
−10.21702140489915585562847361904, −10.21429459004235498656816916133, −9.654284415219037582720301206542, −9.390251081839685706395585325617, −8.510665612064594147786901559654, −8.180833630540973111928522988921, −7.85639046269681270025750241220, −7.16421258992752142002467748583, −6.71686569055411375209606613324, −6.36809279174628169687719555083, −6.28299973055076573697172599269, −5.39677435190006361182924611538, −5.04283324769869745509954827452, −4.22206591085346042349226903022, −4.21509924737431123894137942729, −3.79106148424710745346216876567, −3.03081996372827882612646759374, −2.65123854997023272671893607167, −1.78497635569571174038079952676, −0.76184847244173038283792271090,
0.76184847244173038283792271090, 1.78497635569571174038079952676, 2.65123854997023272671893607167, 3.03081996372827882612646759374, 3.79106148424710745346216876567, 4.21509924737431123894137942729, 4.22206591085346042349226903022, 5.04283324769869745509954827452, 5.39677435190006361182924611538, 6.28299973055076573697172599269, 6.36809279174628169687719555083, 6.71686569055411375209606613324, 7.16421258992752142002467748583, 7.85639046269681270025750241220, 8.180833630540973111928522988921, 8.510665612064594147786901559654, 9.390251081839685706395585325617, 9.654284415219037582720301206542, 10.21429459004235498656816916133, 10.21702140489915585562847361904
