| L(s) = 1 | − 6·3-s + 4·5-s − 20·7-s + 27·9-s + 60·11-s + 26·13-s − 24·15-s − 84·17-s + 60·19-s + 120·21-s − 72·23-s − 210·25-s − 108·27-s + 124·29-s − 108·31-s − 360·33-s − 80·35-s − 36·37-s − 156·39-s − 52·41-s + 32·43-s + 108·45-s − 428·47-s − 134·49-s + 504·51-s − 380·53-s + 240·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.357·5-s − 1.07·7-s + 9-s + 1.64·11-s + 0.554·13-s − 0.413·15-s − 1.19·17-s + 0.724·19-s + 1.24·21-s − 0.652·23-s − 1.67·25-s − 0.769·27-s + 0.794·29-s − 0.625·31-s − 1.89·33-s − 0.386·35-s − 0.159·37-s − 0.640·39-s − 0.198·41-s + 0.113·43-s + 0.357·45-s − 1.32·47-s − 0.390·49-s + 1.38·51-s − 0.984·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 226 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 20 T + 534 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 60 T + 3114 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 84 T + 10582 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 60 T + 6526 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 72 T + 14430 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 124 T - 1586 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 108 T - 16154 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 36 T + 42382 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 52 T + 76666 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 32 T + 150198 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 428 T + 116242 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 380 T + 258142 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1420 T + 882490 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1012 T + 708206 T^{2} + 1012 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 844 T + 778238 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 868 T + 888050 T^{2} + 868 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 60 T + 114886 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 272 T + 993374 T^{2} - 272 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1252 T + 1118698 T^{2} - 1252 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 572 T + 853306 T^{2} - 572 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 708 T + 1762390 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−8.377357846976765520084677095580, −8.045030971501394029901190526248, −7.38638032112470281840826131679, −7.17111346950558968636114282712, −6.51179518081593619516625524139, −6.47740486592304785958528969408, −6.04518496994528146644056450833, −6.01521157931770596600837732441, −5.16668359020135727828426824063, −4.99403943532406969011928591361, −4.34089223235568063759126199697, −3.98385001002060120631303679655, −3.48602073263167174314829164135, −3.35911359350817974734533299611, −2.32615848931917361610287461346, −1.98755785028611166247540976836, −1.34748411718203611169051579098, −0.950807194171427108330702342892, 0, 0,
0.950807194171427108330702342892, 1.34748411718203611169051579098, 1.98755785028611166247540976836, 2.32615848931917361610287461346, 3.35911359350817974734533299611, 3.48602073263167174314829164135, 3.98385001002060120631303679655, 4.34089223235568063759126199697, 4.99403943532406969011928591361, 5.16668359020135727828426824063, 6.01521157931770596600837732441, 6.04518496994528146644056450833, 6.47740486592304785958528969408, 6.51179518081593619516625524139, 7.17111346950558968636114282712, 7.38638032112470281840826131679, 8.045030971501394029901190526248, 8.377357846976765520084677095580
