| L(s) = 1 | − 2-s + 3-s − 2·4-s − 5-s − 6-s + 3·8-s − 4·9-s + 10-s − 2·11-s − 2·12-s + 8·13-s − 15-s + 16-s + 4·17-s + 4·18-s + 19-s + 2·20-s + 2·22-s − 5·23-s + 3·24-s − 8·25-s − 8·26-s − 6·27-s − 5·29-s + 30-s − 5·31-s − 2·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s − 4/3·9-s + 0.316·10-s − 0.603·11-s − 0.577·12-s + 2.21·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.942·18-s + 0.229·19-s + 0.447·20-s + 0.426·22-s − 1.04·23-s + 0.612·24-s − 8/5·25-s − 1.56·26-s − 1.15·27-s − 0.928·29-s + 0.182·30-s − 0.898·31-s − 0.353·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 27 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 51 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7 T + 117 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 19 T + 207 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 127 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 145 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 67 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 147 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 119 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 15 T + 149 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 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
−8.810111054683685287320585521453, −8.761867100193339664683143410959, −8.130358030552861589231581690910, −8.030644668258598871134325168454, −7.60138709349562027439852209703, −7.46153180682731658755418495005, −6.47537611558253441427738058403, −6.11047321348933896266641556122, −5.64829772066498056409156608810, −5.60756832659515713103505028019, −4.87915046895350040837722112097, −4.40828389267130868852530131494, −3.69285470625132819338857745839, −3.52148982483247484138387134864, −3.37068856664004201484573908035, −2.48013895812677977891905219132, −1.75407853264927128953859081447, −1.27367504277911871550885918273, 0, 0,
1.27367504277911871550885918273, 1.75407853264927128953859081447, 2.48013895812677977891905219132, 3.37068856664004201484573908035, 3.52148982483247484138387134864, 3.69285470625132819338857745839, 4.40828389267130868852530131494, 4.87915046895350040837722112097, 5.60756832659515713103505028019, 5.64829772066498056409156608810, 6.11047321348933896266641556122, 6.47537611558253441427738058403, 7.46153180682731658755418495005, 7.60138709349562027439852209703, 8.030644668258598871134325168454, 8.130358030552861589231581690910, 8.761867100193339664683143410959, 8.810111054683685287320585521453
