| L(s) = 1 | − 6·3-s − 10·5-s + 14·7-s + 27·9-s + 92·11-s + 8·13-s + 60·15-s − 44·17-s + 108·19-s − 84·21-s + 320·23-s + 75·25-s − 108·27-s − 236·29-s + 60·31-s − 552·33-s − 140·35-s + 204·37-s − 48·39-s + 44·41-s − 136·43-s − 270·45-s − 400·47-s + 147·49-s + 264·51-s + 16·53-s − 920·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 2.52·11-s + 0.170·13-s + 1.03·15-s − 0.627·17-s + 1.30·19-s − 0.872·21-s + 2.90·23-s + 3/5·25-s − 0.769·27-s − 1.51·29-s + 0.347·31-s − 2.91·33-s − 0.676·35-s + 0.906·37-s − 0.197·39-s + 0.167·41-s − 0.482·43-s − 0.894·45-s − 1.24·47-s + 3/7·49-s + 0.724·51-s + 0.0414·53-s − 2.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.462105641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.462105641\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 92 T + 4758 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T - 2810 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 44 T + 630 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 108 T + 13254 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 320 T + 47934 T^{2} - 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 236 T + 61982 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 60 T + 8462 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 204 T + 108830 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 44 T + 106326 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 136 T + 81718 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 400 T + 106526 T^{2} + 400 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 260838 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 464 T + 458102 T^{2} - 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 684 T + 535646 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 736 T + 602470 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 740 T + 757502 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 424 T + 748558 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T - 143586 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 608 T + 1200710 T^{2} + 608 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1332 T + 1790774 T^{2} + 1332 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2448 T + 3286542 T^{2} + 2448 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−9.177921453757718333950824354576, −8.944169573618173152483225544218, −8.310700904203639243288903286105, −8.090555946886685686017767138206, −7.26621187330938200006448036312, −7.19062689267876026572823402071, −6.84713865495469128093043459020, −6.52118994046565363627019412022, −5.90044820368712555093879712623, −5.56464292668453594584782820939, −4.95664959565074352579971281104, −4.74530689186936749113746434543, −4.08227778400461357996507348673, −4.06264890655981890363827434675, −3.15031012208031401150330143663, −3.06994549216382915173434390907, −1.69752700396306371915130654907, −1.56792883619900974567866060534, −0.791460842089883167524142400397, −0.64852907324223693105868524796,
0.64852907324223693105868524796, 0.791460842089883167524142400397, 1.56792883619900974567866060534, 1.69752700396306371915130654907, 3.06994549216382915173434390907, 3.15031012208031401150330143663, 4.06264890655981890363827434675, 4.08227778400461357996507348673, 4.74530689186936749113746434543, 4.95664959565074352579971281104, 5.56464292668453594584782820939, 5.90044820368712555093879712623, 6.52118994046565363627019412022, 6.84713865495469128093043459020, 7.19062689267876026572823402071, 7.26621187330938200006448036312, 8.090555946886685686017767138206, 8.310700904203639243288903286105, 8.944169573618173152483225544218, 9.177921453757718333950824354576
