| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 7-s + 4·8-s + 3·9-s + 4·10-s + 11-s − 6·12-s + 4·13-s + 2·14-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s − 3·19-s + 6·20-s − 2·21-s + 2·22-s + 7·23-s − 8·24-s + 3·25-s + 8·26-s − 4·27-s + 3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.377·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.301·11-s − 1.73·12-s + 1.10·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 0.688·19-s + 1.34·20-s − 0.436·21-s + 0.426·22-s + 1.45·23-s − 1.63·24-s + 3/5·25-s + 1.56·26-s − 0.769·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.567727113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.567727113\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T - 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 19 T + 192 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 6 T - 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 21 T + 230 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 196 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.58369742668169622881392498137, −10.07896814888613568901950952352, −9.560176306576161195048511404639, −9.135189465923416730781059868944, −8.454113429484776346681778764311, −8.269220544903797731966262923998, −7.27109803947625248603649878617, −7.25349638059761408898454286542, −6.48159766284164875326425405871, −6.36442623568392223854921019561, −5.82657404658869598588534061457, −5.56865023429522812415694669481, −5.06919548701767731828346583917, −4.59322043530120736241495437638, −4.18573751729956530611062536605, −3.69405827722768577241665985395, −2.82179157383162195127400833664, −2.50779933605166344714653458320, −1.35636171358021831321444547883, −1.19219428858697637835555291853,
1.19219428858697637835555291853, 1.35636171358021831321444547883, 2.50779933605166344714653458320, 2.82179157383162195127400833664, 3.69405827722768577241665985395, 4.18573751729956530611062536605, 4.59322043530120736241495437638, 5.06919548701767731828346583917, 5.56865023429522812415694669481, 5.82657404658869598588534061457, 6.36442623568392223854921019561, 6.48159766284164875326425405871, 7.25349638059761408898454286542, 7.27109803947625248603649878617, 8.269220544903797731966262923998, 8.454113429484776346681778764311, 9.135189465923416730781059868944, 9.560176306576161195048511404639, 10.07896814888613568901950952352, 10.58369742668169622881392498137
