| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 3·9-s − 5·11-s + 13-s − 14-s − 16-s + 5·17-s + 3·18-s − 4·19-s + 5·22-s − 3·23-s − 5·25-s − 26-s − 28-s − 4·29-s − 5·31-s − 5·32-s − 5·34-s + 3·36-s + 9·37-s + 4·38-s − 7·41-s + 3·43-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 9-s − 1.50·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s + 1.21·17-s + 0.707·18-s − 0.917·19-s + 1.06·22-s − 0.625·23-s − 25-s − 0.196·26-s − 0.188·28-s − 0.742·29-s − 0.898·31-s − 0.883·32-s − 0.857·34-s + 1/2·36-s + 1.47·37-s + 0.648·38-s − 1.09·41-s + 0.457·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 7 | $C_4$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 19 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 91 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 61 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T - 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T - 17 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 214 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−15.5431657874, −14.7844473315, −14.5767734977, −14.1325888644, −13.4464361926, −13.2732752281, −12.6906797513, −12.2045515934, −11.4595293433, −11.1411267054, −10.6006366347, −10.1228839498, −9.73557097784, −9.09529841462, −8.49300235635, −8.20129435365, −7.55748478798, −7.46021660361, −6.11947805636, −5.74606204821, −5.19693385719, −4.46204949162, −3.69652055447, −2.78176727319, −1.76136175816, 0,
1.76136175816, 2.78176727319, 3.69652055447, 4.46204949162, 5.19693385719, 5.74606204821, 6.11947805636, 7.46021660361, 7.55748478798, 8.20129435365, 8.49300235635, 9.09529841462, 9.73557097784, 10.1228839498, 10.6006366347, 11.1411267054, 11.4595293433, 12.2045515934, 12.6906797513, 13.2732752281, 13.4464361926, 14.1325888644, 14.5767734977, 14.7844473315, 15.5431657874
