| L(s) = 1 | − 2·5-s − 2·7-s − 6·9-s + 2·11-s − 4·19-s − 4·23-s + 3·25-s + 4·31-s + 4·35-s − 4·41-s − 8·43-s + 12·45-s − 4·47-s + 3·49-s + 8·53-s − 4·55-s − 12·59-s + 12·63-s − 8·67-s + 24·73-s − 4·77-s − 12·79-s + 27·81-s + 12·89-s + 8·95-s − 12·99-s + 8·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 2·9-s + 0.603·11-s − 0.917·19-s − 0.834·23-s + 3/5·25-s + 0.718·31-s + 0.676·35-s − 0.624·41-s − 1.21·43-s + 1.78·45-s − 0.583·47-s + 3/7·49-s + 1.09·53-s − 0.539·55-s − 1.56·59-s + 1.51·63-s − 0.977·67-s + 2.80·73-s − 0.455·77-s − 1.35·79-s + 3·81-s + 1.27·89-s + 0.820·95-s − 1.20·99-s + 0.796·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37945600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.071337988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071337988\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 24 T + 270 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_4$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
| 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
−8.217931200167430154203903257599, −8.094344028898204548724967188905, −7.47841881979692677070518554860, −7.28239852196738513151909937024, −6.60762088260732386871389365416, −6.52785138149006083098260652526, −6.11773481530950789819103964463, −5.91378985305989834853973222072, −5.39551449022148277519209761652, −5.01116371461272293142368558002, −4.54577272200906053633414759427, −4.27624186926950911299816675905, −3.58842321752206696517292044342, −3.50863487871347615927752887624, −3.04588413031692263863172378566, −2.74118348550971981111641856736, −2.04828603202633192540972281833, −1.78317364699155316393025354613, −0.64694323777064887843649271325, −0.41016070304102084483541937010,
0.41016070304102084483541937010, 0.64694323777064887843649271325, 1.78317364699155316393025354613, 2.04828603202633192540972281833, 2.74118348550971981111641856736, 3.04588413031692263863172378566, 3.50863487871347615927752887624, 3.58842321752206696517292044342, 4.27624186926950911299816675905, 4.54577272200906053633414759427, 5.01116371461272293142368558002, 5.39551449022148277519209761652, 5.91378985305989834853973222072, 6.11773481530950789819103964463, 6.52785138149006083098260652526, 6.60762088260732386871389365416, 7.28239852196738513151909937024, 7.47841881979692677070518554860, 8.094344028898204548724967188905, 8.217931200167430154203903257599
