| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 5-s − 4·6-s + 2·7-s + 4·8-s + 3·9-s − 2·10-s + 5·11-s − 6·12-s + 2·13-s + 4·14-s + 2·15-s + 5·16-s + 17-s + 6·18-s + 5·19-s − 3·20-s − 4·21-s + 10·22-s + 3·23-s − 8·24-s + 25-s + 4·26-s − 4·27-s + 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.447·5-s − 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 0.632·10-s + 1.50·11-s − 1.73·12-s + 0.554·13-s + 1.06·14-s + 0.516·15-s + 5/4·16-s + 0.242·17-s + 1.41·18-s + 1.14·19-s − 0.670·20-s − 0.872·21-s + 2.13·22-s + 0.625·23-s − 1.63·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.825009045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.825009045\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 15 T + 120 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 156 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−11.43483643089544292602743656076, −10.95902301042485306900855631394, −10.15243844260323863445322529913, −10.09375930711130694623145937064, −9.220752912421352144473568592103, −8.953394879882585085534439410086, −8.021259355367973261734314365488, −7.81391489647087793893531511872, −7.19826094541585290985540074696, −6.77192914266805202384889861604, −6.17195708811421168737571013421, −6.11387749439743136317593171945, −5.23069746258966728509319936006, −5.03677385107277837904297481321, −4.41834128660271042071707525915, −4.02564892686347259075982140467, −3.44521090361969176928239126777, −2.81767171958026255079962866204, −1.56033051386842278131548780488, −1.18483450496188729564153348359,
1.18483450496188729564153348359, 1.56033051386842278131548780488, 2.81767171958026255079962866204, 3.44521090361969176928239126777, 4.02564892686347259075982140467, 4.41834128660271042071707525915, 5.03677385107277837904297481321, 5.23069746258966728509319936006, 6.11387749439743136317593171945, 6.17195708811421168737571013421, 6.77192914266805202384889861604, 7.19826094541585290985540074696, 7.81391489647087793893531511872, 8.021259355367973261734314365488, 8.953394879882585085534439410086, 9.220752912421352144473568592103, 10.09375930711130694623145937064, 10.15243844260323863445322529913, 10.95902301042485306900855631394, 11.43483643089544292602743656076
