| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 2·7-s − 4·8-s + 3·9-s − 4·10-s − 2·11-s + 6·12-s + 2·13-s + 4·14-s + 4·15-s + 5·16-s + 4·17-s − 6·18-s + 6·20-s − 4·21-s + 4·22-s + 2·23-s − 8·24-s + 3·25-s − 4·26-s + 4·27-s − 6·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s + 1.73·12-s + 0.554·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s − 1.41·18-s + 1.34·20-s − 0.872·21-s + 0.852·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s − 1.13·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.354905114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.354905114\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 90 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 22 T + 250 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 106 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.434155870949413074614177549538, −8.287588536890770274413637304999, −7.73085484875391877877317537249, −7.56708274635814454890847957683, −7.13880802825351487450414238354, −6.86884969193298084274851577905, −6.31034354799074678245610747258, −6.12974142443436564880072227242, −5.57104383784098561093169610635, −5.50674131385642535565527901560, −4.53114614787236694140449008053, −4.49763089799424932740494120511, −3.55450675398024210235827571674, −3.38581729956274980599513608993, −2.90136211314482545414220723143, −2.62284177555990265232042361189, −2.02683967629609381229260916704, −1.78898968673255261136769108942, −0.891540137263366751752946541433, −0.74883801951199862721704159242,
0.74883801951199862721704159242, 0.891540137263366751752946541433, 1.78898968673255261136769108942, 2.02683967629609381229260916704, 2.62284177555990265232042361189, 2.90136211314482545414220723143, 3.38581729956274980599513608993, 3.55450675398024210235827571674, 4.49763089799424932740494120511, 4.53114614787236694140449008053, 5.50674131385642535565527901560, 5.57104383784098561093169610635, 6.12974142443436564880072227242, 6.31034354799074678245610747258, 6.86884969193298084274851577905, 7.13880802825351487450414238354, 7.56708274635814454890847957683, 7.73085484875391877877317537249, 8.287588536890770274413637304999, 8.434155870949413074614177549538
