L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 3·5-s − 6·6-s − 4·8-s + 6·9-s + 6·10-s + 6·11-s + 9·12-s + 2·13-s − 9·15-s + 5·16-s + 6·17-s − 12·18-s − 7·19-s − 9·20-s − 12·22-s − 3·23-s − 12·24-s + 5·25-s − 4·26-s + 9·27-s − 6·29-s + 18·30-s − 4·31-s − 6·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 1.34·5-s − 2.44·6-s − 1.41·8-s + 2·9-s + 1.89·10-s + 1.80·11-s + 2.59·12-s + 0.554·13-s − 2.32·15-s + 5/4·16-s + 1.45·17-s − 2.82·18-s − 1.60·19-s − 2.01·20-s − 2.55·22-s − 0.625·23-s − 2.44·24-s + 25-s − 0.784·26-s + 1.73·27-s − 1.11·29-s + 3.28·30-s − 0.718·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.683159637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.683159637\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.30342881602350466629626547855, −9.580943614488192436286195946268, −9.378211750079020802052420405131, −9.085960523225293808189055890237, −8.635880852005952218943819256235, −8.283882833156454088544932815282, −7.87444730208417311617148945876, −7.76210689350065334241214993074, −7.20001285286957715535749432011, −6.68607737276687456952568019909, −6.43306271838766420082089731189, −5.78024673367671909930095944143, −4.89829546917896514722514151712, −3.99602832788807490235429905619, −3.89318796247202082018883405276, −3.48347600463853945544674083816, −2.95957098769143775327443412310, −1.90123015979025708737691624133, −1.75456471170968396787413712455, −0.73356393598858232553921223617,
0.73356393598858232553921223617, 1.75456471170968396787413712455, 1.90123015979025708737691624133, 2.95957098769143775327443412310, 3.48347600463853945544674083816, 3.89318796247202082018883405276, 3.99602832788807490235429905619, 4.89829546917896514722514151712, 5.78024673367671909930095944143, 6.43306271838766420082089731189, 6.68607737276687456952568019909, 7.20001285286957715535749432011, 7.76210689350065334241214993074, 7.87444730208417311617148945876, 8.283882833156454088544932815282, 8.635880852005952218943819256235, 9.085960523225293808189055890237, 9.378211750079020802052420405131, 9.580943614488192436286195946268, 10.30342881602350466629626547855