| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s + 2·5-s − 6·6-s + 3·7-s − 4·8-s + 4·9-s − 4·10-s − 7·11-s + 9·12-s + 3·13-s − 6·14-s + 6·15-s + 5·16-s + 3·17-s − 8·18-s + 19-s + 6·20-s + 9·21-s + 14·22-s − 2·23-s − 12·24-s + 3·25-s − 6·26-s + 6·27-s + 9·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 2.44·6-s + 1.13·7-s − 1.41·8-s + 4/3·9-s − 1.26·10-s − 2.11·11-s + 2.59·12-s + 0.832·13-s − 1.60·14-s + 1.54·15-s + 5/4·16-s + 0.727·17-s − 1.88·18-s + 0.229·19-s + 1.34·20-s + 1.96·21-s + 2.98·22-s − 0.417·23-s − 2.44·24-s + 3/5·25-s − 1.17·26-s + 1.15·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598771131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598771131\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−12.54562329491617748097097436723, −11.75744974693688806614818154203, −11.28144094921353663359496219197, −10.71046230091007479456466608837, −10.29941953718408687215638141424, −10.06568592678611651751133565890, −9.258305412585825402855487750376, −9.132013969731634669302277945894, −8.341857495284555451505842873076, −8.194436290827410356788454580987, −7.67615958829551750295562810625, −7.57180952442899203360257444861, −6.49361484218236037802429356517, −5.91211893904072091645324921101, −5.22867063412821604665322838870, −4.55549946053216157588369309046, −3.18920512476946472385632829776, −2.91208024671020648339608776976, −2.11808372085269199478702255532, −1.46438094505397839462929047850,
1.46438094505397839462929047850, 2.11808372085269199478702255532, 2.91208024671020648339608776976, 3.18920512476946472385632829776, 4.55549946053216157588369309046, 5.22867063412821604665322838870, 5.91211893904072091645324921101, 6.49361484218236037802429356517, 7.57180952442899203360257444861, 7.67615958829551750295562810625, 8.194436290827410356788454580987, 8.341857495284555451505842873076, 9.132013969731634669302277945894, 9.258305412585825402855487750376, 10.06568592678611651751133565890, 10.29941953718408687215638141424, 10.71046230091007479456466608837, 11.28144094921353663359496219197, 11.75744974693688806614818154203, 12.54562329491617748097097436723
