| L(s) = 1 | − 3·2-s + 2·4-s − 4·5-s + 10·7-s + 3·8-s + 12·10-s + 8·11-s + 21·13-s − 30·14-s − 3·16-s + 8·17-s + 38·19-s − 8·20-s − 24·22-s − 4·23-s + 25·25-s − 63·26-s + 20·28-s + 96·29-s + 12·32-s − 24·34-s − 40·35-s − 114·38-s − 12·40-s + 24·41-s − 11·43-s + 16·44-s + ⋯ |
| L(s) = 1 | − 3/2·2-s + 1/2·4-s − 4/5·5-s + 10/7·7-s + 3/8·8-s + 6/5·10-s + 8/11·11-s + 1.61·13-s − 2.14·14-s − 0.187·16-s + 8/17·17-s + 2·19-s − 2/5·20-s − 1.09·22-s − 0.173·23-s + 25-s − 2.42·26-s + 5/7·28-s + 3.31·29-s + 3/8·32-s − 0.705·34-s − 8/7·35-s − 3·38-s − 0.299·40-s + 0.585·41-s − 0.255·43-s + 4/11·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9777139589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9777139589\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 9 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 8 T - 225 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 513 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 96 T + 3913 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1895 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2495 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 24 T + 1873 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T - 1728 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T + 927 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 84 T + 5161 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 84 T + 5833 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 79 T + 2520 T^{2} - 79 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 33 T + 4852 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 192 T + 17329 T^{2} + 192 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 5328 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 57 T + 7324 T^{2} - 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 112 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 132 T + 13729 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 192 T + 21697 T^{2} - 192 p^{2} T^{3} + p^{4} T^{4} \) |
| 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
−12.44291915001688594395832842498, −11.99808360676178173231373327820, −11.71159641362261434318355240137, −11.26266995199753655785954059922, −10.53448663545078652314528751172, −10.43905137820371939166026015743, −9.534789609545284517687709110933, −9.178905334659953993992449511860, −8.497362154712958043535128993308, −8.430065414635323508633932477008, −7.71651997191420605503174514990, −7.57552116956633540306848179266, −6.50474331506494351935680482353, −6.08275203646606437763947635605, −4.87582581622350282015302547119, −4.72887404246960375159461306134, −3.69068555548219883249910035557, −2.98966419844780302698148047961, −1.15898111851617525717697489685, −1.11150846709310020999855785921,
1.11150846709310020999855785921, 1.15898111851617525717697489685, 2.98966419844780302698148047961, 3.69068555548219883249910035557, 4.72887404246960375159461306134, 4.87582581622350282015302547119, 6.08275203646606437763947635605, 6.50474331506494351935680482353, 7.57552116956633540306848179266, 7.71651997191420605503174514990, 8.430065414635323508633932477008, 8.497362154712958043535128993308, 9.178905334659953993992449511860, 9.534789609545284517687709110933, 10.43905137820371939166026015743, 10.53448663545078652314528751172, 11.26266995199753655785954059922, 11.71159641362261434318355240137, 11.99808360676178173231373327820, 12.44291915001688594395832842498
