| L(s) = 1 | − 2·2-s + 3·4-s + 3·5-s + 2·7-s − 4·8-s − 6·10-s + 3·11-s + 4·13-s − 4·14-s + 5·16-s − 3·17-s + 10·19-s + 9·20-s − 6·22-s − 9·23-s + 5·25-s − 8·26-s + 6·28-s − 6·29-s + 4·31-s − 6·32-s + 6·34-s + 6·35-s + 4·37-s − 20·38-s − 12·40-s + 15·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.34·5-s + 0.755·7-s − 1.41·8-s − 1.89·10-s + 0.904·11-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.727·17-s + 2.29·19-s + 2.01·20-s − 1.27·22-s − 1.87·23-s + 25-s − 1.56·26-s + 1.13·28-s − 1.11·29-s + 0.718·31-s − 1.06·32-s + 1.02·34-s + 1.01·35-s + 0.657·37-s − 3.24·38-s − 1.89·40-s + 2.34·41-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
| \(L(1)\) |
\(\approx\) |
\(2.093137985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093137985\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 15 T + 130 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.611426880671117014573603530527, −9.553029099132009361883548736634, −9.374193050761032141502780033119, −9.052040037772952561483046744056, −8.277735268974693200492175365259, −8.155637111832045415082538407090, −7.55487777992610276284619447478, −7.42458180415379935621344317395, −6.56715271469849864491990788155, −6.36549820293758308041327942502, −5.83861194238455432709651689682, −5.72514461951518833019983702340, −4.98470719014140810015651110823, −4.38169554532108659490936761861, −3.69745885480400443806823455745, −3.24589792205901543785445276530, −2.23144714084569893290900727542, −2.13628201850153038060075765431, −1.28480170843469111711011416771, −0.938150861051641907853982587863,
0.938150861051641907853982587863, 1.28480170843469111711011416771, 2.13628201850153038060075765431, 2.23144714084569893290900727542, 3.24589792205901543785445276530, 3.69745885480400443806823455745, 4.38169554532108659490936761861, 4.98470719014140810015651110823, 5.72514461951518833019983702340, 5.83861194238455432709651689682, 6.36549820293758308041327942502, 6.56715271469849864491990788155, 7.42458180415379935621344317395, 7.55487777992610276284619447478, 8.155637111832045415082538407090, 8.277735268974693200492175365259, 9.052040037772952561483046744056, 9.374193050761032141502780033119, 9.553029099132009361883548736634, 9.611426880671117014573603530527
