| L(s) = 1 | − 4·2-s + 12·4-s − 8·5-s − 32·8-s + 32·10-s + 10·13-s + 80·16-s + 16·17-s − 96·20-s + 25·25-s − 40·26-s + 40·29-s − 192·32-s − 64·34-s + 70·37-s + 256·40-s + 160·41-s + 98·49-s − 100·50-s + 120·52-s + 112·53-s − 160·58-s + 22·61-s + 448·64-s − 80·65-s + 192·68-s − 110·73-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 8/5·5-s − 4·8-s + 16/5·10-s + 0.769·13-s + 5·16-s + 0.941·17-s − 4.79·20-s + 25-s − 1.53·26-s + 1.37·29-s − 6·32-s − 1.88·34-s + 1.89·37-s + 32/5·40-s + 3.90·41-s + 2·49-s − 2·50-s + 2.30·52-s + 2.11·53-s − 2.75·58-s + 0.360·61-s + 7·64-s − 1.23·65-s + 2.82·68-s − 1.50·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7309711401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7309711401\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 69 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 16 T - 33 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40 T + 759 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 70 T + 3531 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 22 T - 3237 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 110 T + 6771 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 160 T + 17679 T^{2} - 160 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
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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
−11.42314697617544682812712318328, −11.10034675815188262493856701782, −10.53712746844560854557603334554, −10.43621669501808987442458672724, −9.519334601682887062453703182589, −9.385248864412942152020010196831, −8.659277787449480728695820620469, −8.380697263077570212440411947796, −7.81949128746743947609387918957, −7.58691665765103798899556761540, −7.19751639288390397635840454626, −6.51780485172428134108876869147, −5.92074395385771051389233910138, −5.53971010336517250355527958985, −4.08003335280628197559644753976, −4.01028041070705099324753152079, −2.84801272538593139837347338514, −2.56345218017214313659615948721, −1.06193326555299248479842619375, −0.72084931986091153669628774411,
0.72084931986091153669628774411, 1.06193326555299248479842619375, 2.56345218017214313659615948721, 2.84801272538593139837347338514, 4.01028041070705099324753152079, 4.08003335280628197559644753976, 5.53971010336517250355527958985, 5.92074395385771051389233910138, 6.51780485172428134108876869147, 7.19751639288390397635840454626, 7.58691665765103798899556761540, 7.81949128746743947609387918957, 8.380697263077570212440411947796, 8.659277787449480728695820620469, 9.385248864412942152020010196831, 9.519334601682887062453703182589, 10.43621669501808987442458672724, 10.53712746844560854557603334554, 11.10034675815188262493856701782, 11.42314697617544682812712318328
