| L(s) = 1 | − 4·3-s − 8·5-s − 4·7-s + 10·9-s − 4·11-s + 32·15-s − 12·19-s + 16·21-s − 12·23-s + 30·25-s − 20·27-s − 8·29-s − 16·31-s + 16·33-s + 32·35-s + 12·41-s − 12·43-s − 80·45-s + 8·49-s − 16·53-s + 32·55-s + 48·57-s + 4·59-s − 40·63-s − 12·67-s + 48·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 3.57·5-s − 1.51·7-s + 10/3·9-s − 1.20·11-s + 8.26·15-s − 2.75·19-s + 3.49·21-s − 2.50·23-s + 6·25-s − 3.84·27-s − 1.48·29-s − 2.87·31-s + 2.78·33-s + 5.40·35-s + 1.87·41-s − 1.82·43-s − 11.9·45-s + 8/7·49-s − 2.19·53-s + 4.31·55-s + 6.35·57-s + 0.520·59-s − 5.03·63-s − 1.46·67-s + 5.77·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 48 T^{3} + 2 T^{4} - 48 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 492 T^{3} + 2402 T^{4} + 492 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 300 T^{3} + 1246 T^{4} + 300 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 34 T^{2} + 200 T^{3} + 1026 T^{4} + 200 p T^{5} + 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 144 T^{3} + 2 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 516 T^{3} + 3694 T^{4} - 516 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 564 T^{3} - 6334 T^{4} - 564 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} - 128 T^{3} - 3294 T^{4} - 128 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 668 T^{3} - 2718 T^{4} + 668 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 240 T^{3} + 2 T^{4} - 240 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 876 T^{3} + 7010 T^{4} + 876 p T^{5} + 86 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 876 T^{3} + 10654 T^{4} - 876 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} - 1052 T^{3} - 4254 T^{4} - 1052 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 516 T^{3} + 1582 T^{4} + 516 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.369288240211191731804004718554, −7.87992134849721238846432543250, −7.73638679902001296997842820626, −7.55303383848477720304195090525, −7.54071892197507665960018387457, −6.97053122671937269980499394010, −6.96530411127897513533102201601, −6.87588371295695491893327934543, −6.32444256351903877454830991580, −6.07768546722516718535178817222, −5.85035583164364066053543868256, −5.73276398858471881777660743342, −5.70164067647619500056107864730, −4.87159611387599575493655641723, −4.74682585856384818012356174781, −4.67976371386297070945644902188, −4.28851912548652581209807242322, −3.81970542588647787434158516843, −3.79840407367954204567949471673, −3.71029311592720875191203583533, −3.56197289972756328074027556091, −2.64383555011288117171709345271, −2.60529126452859993621792209197, −1.83141158067301700284507424761, −1.47767808027432794889727811471, 0, 0, 0, 0,
1.47767808027432794889727811471, 1.83141158067301700284507424761, 2.60529126452859993621792209197, 2.64383555011288117171709345271, 3.56197289972756328074027556091, 3.71029311592720875191203583533, 3.79840407367954204567949471673, 3.81970542588647787434158516843, 4.28851912548652581209807242322, 4.67976371386297070945644902188, 4.74682585856384818012356174781, 4.87159611387599575493655641723, 5.70164067647619500056107864730, 5.73276398858471881777660743342, 5.85035583164364066053543868256, 6.07768546722516718535178817222, 6.32444256351903877454830991580, 6.87588371295695491893327934543, 6.96530411127897513533102201601, 6.97053122671937269980499394010, 7.54071892197507665960018387457, 7.55303383848477720304195090525, 7.73638679902001296997842820626, 7.87992134849721238846432543250, 8.369288240211191731804004718554
