| L(s) = 1 | − 2-s + 3·5-s − 7-s + 8-s − 3·10-s + 6·11-s + 13-s + 14-s − 16-s + 6·17-s + 4·19-s − 6·22-s + 6·23-s + 5·25-s − 26-s + 9·29-s + 10·31-s − 6·34-s − 3·35-s − 14·37-s − 4·38-s + 3·40-s − 6·41-s + 4·43-s − 6·46-s − 6·47-s − 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s − 0.377·7-s + 0.353·8-s − 0.948·10-s + 1.80·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.27·22-s + 1.25·23-s + 25-s − 0.196·26-s + 1.67·29-s + 1.79·31-s − 1.02·34-s − 0.507·35-s − 2.30·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 0.884·46-s − 0.875·47-s − 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.584771993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.584771993\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 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 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| 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
−9.959412636747884270068846904767, −9.793009425611540517045828265720, −9.102048515020655962584395839390, −8.996558239787216145534504043210, −8.456095146412158674561356909972, −8.259221759748213239029838352112, −7.38635002088883832413786120331, −7.21732197553549228913214154370, −6.52971027584129180400570239012, −6.34165974762890287419742700325, −6.06612286073431944210998771369, −5.23040478913913337874608773361, −5.00854605935149852447247633123, −4.53479084820914783957455729668, −3.62751817610428634708668978409, −3.27536199925732988859333826106, −2.85175169230445470170579818927, −1.87430426293817310268732094758, −1.25317574174354436405884938233, −0.997023425202170479832554010303,
0.997023425202170479832554010303, 1.25317574174354436405884938233, 1.87430426293817310268732094758, 2.85175169230445470170579818927, 3.27536199925732988859333826106, 3.62751817610428634708668978409, 4.53479084820914783957455729668, 5.00854605935149852447247633123, 5.23040478913913337874608773361, 6.06612286073431944210998771369, 6.34165974762890287419742700325, 6.52971027584129180400570239012, 7.21732197553549228913214154370, 7.38635002088883832413786120331, 8.259221759748213239029838352112, 8.456095146412158674561356909972, 8.996558239787216145534504043210, 9.102048515020655962584395839390, 9.793009425611540517045828265720, 9.959412636747884270068846904767
