| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 4·7-s − 10-s − 2·11-s − 4·14-s + 15-s + 17-s + 4·21-s − 2·22-s + 29-s + 30-s − 32-s + 2·33-s + 34-s + 4·35-s + 4·42-s + 43-s − 47-s + 10·49-s − 51-s + 2·55-s + 58-s − 2·61-s − 64-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 4·7-s − 10-s − 2·11-s − 4·14-s + 15-s + 17-s + 4·21-s − 2·22-s + 29-s + 30-s − 32-s + 2·33-s + 34-s + 4·35-s + 4·42-s + 43-s − 47-s + 10·49-s − 51-s + 2·55-s + 58-s − 2·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1454660510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1454660510\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 47 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 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
−10.70955213756479811362162784734, −10.66530401221304054625069977646, −10.03621006059901212186679389412, −9.903835962416619268574009052026, −9.193104337629727275177025343924, −9.138586549043879475055551401652, −8.017543413469060266259296276215, −7.907142287776799694358110661311, −7.16341376048102623222240477469, −6.81704586177366476767873952453, −6.40523725612466881981559384236, −5.86092356609639097778967527964, −5.54299563948557272267303212815, −5.25793040432952100231640176051, −4.32172605841167439211353665450, −3.99711573646764827778492321787, −3.35097268936834539866608727605, −2.97583841927247291517852579278, −2.68994276532299978792457685288, −0.36271213619496852354021242570,
0.36271213619496852354021242570, 2.68994276532299978792457685288, 2.97583841927247291517852579278, 3.35097268936834539866608727605, 3.99711573646764827778492321787, 4.32172605841167439211353665450, 5.25793040432952100231640176051, 5.54299563948557272267303212815, 5.86092356609639097778967527964, 6.40523725612466881981559384236, 6.81704586177366476767873952453, 7.16341376048102623222240477469, 7.907142287776799694358110661311, 8.017543413469060266259296276215, 9.138586549043879475055551401652, 9.193104337629727275177025343924, 9.903835962416619268574009052026, 10.03621006059901212186679389412, 10.66530401221304054625069977646, 10.70955213756479811362162784734
