| L(s) = 1 | − 2·2-s + 2·4-s − 6·5-s − 4·8-s + 3·9-s + 12·10-s + 8·16-s − 6·17-s − 6·18-s − 12·20-s + 11·25-s + 16·29-s − 8·32-s + 12·34-s + 6·36-s − 6·37-s + 24·40-s − 18·45-s − 2·49-s − 22·50-s + 2·53-s − 32·58-s − 18·61-s + 8·64-s − 12·68-s − 12·72-s + 30·73-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 2.68·5-s − 1.41·8-s + 9-s + 3.79·10-s + 2·16-s − 1.45·17-s − 1.41·18-s − 2.68·20-s + 11/5·25-s + 2.97·29-s − 1.41·32-s + 2.05·34-s + 36-s − 0.986·37-s + 3.79·40-s − 2.68·45-s − 2/7·49-s − 3.11·50-s + 0.274·53-s − 4.20·58-s − 2.30·61-s + 64-s − 1.45·68-s − 1.41·72-s + 3.51·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09653238649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09653238649\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 19 T^{2} - 1848 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 27 T + 332 T^{2} - 27 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−13.13342652700473590833995356274, −12.43600940000912869275370860664, −12.15519613932422405978079593033, −12.13207478256375503000632186598, −11.76458414434093393874729517241, −11.67827119083028944990220808552, −10.81906248416036932328431871622, −10.75381845164304346558910523698, −10.58044058794066070153863344420, −10.02830839705813508541350777870, −9.500752157084911352039298183007, −9.085135028380696945126021414053, −9.025421267076234067385984904885, −8.366268142254710935803300809034, −7.948113451221559700796396614690, −7.85822589789037405549931951894, −7.74188899548957632199770281072, −6.70339154631673365237492672570, −6.68917178271240966326952511997, −6.41992193808807162362530336494, −5.21055972755172199076251286197, −4.72839043081946664752190802459, −3.97935651422220664617584685071, −3.79023935525388777841933541037, −2.79175426214166254513393836923,
2.79175426214166254513393836923, 3.79023935525388777841933541037, 3.97935651422220664617584685071, 4.72839043081946664752190802459, 5.21055972755172199076251286197, 6.41992193808807162362530336494, 6.68917178271240966326952511997, 6.70339154631673365237492672570, 7.74188899548957632199770281072, 7.85822589789037405549931951894, 7.948113451221559700796396614690, 8.366268142254710935803300809034, 9.025421267076234067385984904885, 9.085135028380696945126021414053, 9.500752157084911352039298183007, 10.02830839705813508541350777870, 10.58044058794066070153863344420, 10.75381845164304346558910523698, 10.81906248416036932328431871622, 11.67827119083028944990220808552, 11.76458414434093393874729517241, 12.13207478256375503000632186598, 12.15519613932422405978079593033, 12.43600940000912869275370860664, 13.13342652700473590833995356274
