| L(s) = 1 | + 2-s + 2·3-s − 2·4-s + 2·6-s + 2·7-s − 3·8-s − 3·9-s − 4·12-s + 4·13-s + 2·14-s + 16-s − 3·18-s − 10·19-s + 4·21-s + 2·23-s − 6·24-s + 4·26-s − 14·27-s − 4·28-s + 6·29-s − 18·31-s + 2·32-s + 6·36-s − 2·37-s − 10·38-s + 8·39-s + 4·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 4-s + 0.816·6-s + 0.755·7-s − 1.06·8-s − 9-s − 1.15·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.707·18-s − 2.29·19-s + 0.872·21-s + 0.417·23-s − 1.22·24-s + 0.784·26-s − 2.69·27-s − 0.755·28-s + 1.11·29-s − 3.23·31-s + 0.353·32-s + 36-s − 0.328·37-s − 1.62·38-s + 1.28·39-s + 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16200625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16200625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 77 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 201 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 101 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 178 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.350329933734952409859327177506, −8.125933760229292172296227731292, −7.59722754166431819530618122077, −7.39590964487500001541968724071, −6.70549368654592222642490328159, −6.23876918012109201011593600370, −5.86541483197722003307724598324, −5.77890108494033546614824547625, −4.98968771633073778616943756383, −4.98868513729720291122205823140, −4.31628866076734236043525363972, −4.08238891614911344155956974363, −3.61938829907485201539472834940, −3.30668076776429185708201866173, −2.89752439716216048790040016075, −2.32208641849278377881195589094, −1.82352050973926868699688820598, −1.39833687032560782469157297142, 0, 0,
1.39833687032560782469157297142, 1.82352050973926868699688820598, 2.32208641849278377881195589094, 2.89752439716216048790040016075, 3.30668076776429185708201866173, 3.61938829907485201539472834940, 4.08238891614911344155956974363, 4.31628866076734236043525363972, 4.98868513729720291122205823140, 4.98968771633073778616943756383, 5.77890108494033546614824547625, 5.86541483197722003307724598324, 6.23876918012109201011593600370, 6.70549368654592222642490328159, 7.39590964487500001541968724071, 7.59722754166431819530618122077, 8.125933760229292172296227731292, 8.350329933734952409859327177506
