| L(s) = 1 | − 14·5-s + 24·7-s + 28·11-s + 74·13-s + 164·17-s + 184·19-s − 8·23-s + 125·25-s + 138·29-s − 80·31-s − 336·35-s + 60·37-s − 282·41-s − 4·43-s − 240·47-s + 343·49-s − 260·53-s − 392·55-s − 596·59-s + 218·61-s − 1.03e3·65-s + 436·67-s + 1.71e3·71-s − 1.99e3·73-s + 672·77-s + 32·79-s + 1.50e3·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s + 1.29·7-s + 0.767·11-s + 1.57·13-s + 2.33·17-s + 2.22·19-s − 0.0725·23-s + 25-s + 0.883·29-s − 0.463·31-s − 1.62·35-s + 0.266·37-s − 1.07·41-s − 0.0141·43-s − 0.744·47-s + 49-s − 0.673·53-s − 0.961·55-s − 1.31·59-s + 0.457·61-s − 1.97·65-s + 0.795·67-s + 2.86·71-s − 3.20·73-s + 0.994·77-s + 0.0455·79-s + 1.99·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.303224949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.303224949\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 14 T + 71 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 24 T + 233 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 28 T - 547 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 74 T + 3279 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T - 12103 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 138 T - 5345 T^{2} - 138 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 80 T - 23391 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 282 T + 10603 T^{2} + 282 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 79491 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 240 T - 46223 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 130 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 596 T + 149837 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 218 T - 179457 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 436 T - 110667 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 856 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 998 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 32 T - 492015 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1508 T + 1702277 T^{2} - 1508 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 246 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 866 T - 162717 T^{2} + 866 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48023480195193242931742421118, −9.819928736160054871634982316948, −9.625239550506241337803110864179, −8.961707295822698353477469615498, −8.468341391223881290872138516050, −8.126658211513499946455287313376, −7.80767048757835969635895123831, −7.49368367968520898836874632575, −6.96611293003743937221993075798, −6.34509132500831454100560780044, −5.81264931866387607073584964152, −5.14451637932726973327170862026, −5.05878181254525299583969326866, −4.24040063107997487079823841393, −3.55875316649750929410341632960, −3.51578422186503612533377845469, −2.86386113368538552078314028293, −1.45901289996908664180369986921, −1.32102376172182133613646497366, −0.71017897440555711550347109691,
0.71017897440555711550347109691, 1.32102376172182133613646497366, 1.45901289996908664180369986921, 2.86386113368538552078314028293, 3.51578422186503612533377845469, 3.55875316649750929410341632960, 4.24040063107997487079823841393, 5.05878181254525299583969326866, 5.14451637932726973327170862026, 5.81264931866387607073584964152, 6.34509132500831454100560780044, 6.96611293003743937221993075798, 7.49368367968520898836874632575, 7.80767048757835969635895123831, 8.126658211513499946455287313376, 8.468341391223881290872138516050, 8.961707295822698353477469615498, 9.625239550506241337803110864179, 9.819928736160054871634982316948, 10.48023480195193242931742421118
