| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 25·5-s + 36·6-s + 42.1·7-s − 64·8-s + 81·9-s + 100·10-s − 210.·11-s − 144·12-s − 674.·13-s − 168.·14-s + 225·15-s + 256·16-s − 1.99e3·17-s − 324·18-s + 361·19-s − 400·20-s − 379.·21-s + 843.·22-s + 2.91e3·23-s + 576·24-s + 625·25-s + 2.69e3·26-s − 729·27-s + 674.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.325·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.525·11-s − 0.288·12-s − 1.10·13-s − 0.229·14-s + 0.258·15-s + 0.250·16-s − 1.67·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.187·21-s + 0.371·22-s + 1.14·23-s + 0.204·24-s + 0.200·25-s + 0.782·26-s − 0.192·27-s + 0.162·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5060308873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5060308873\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 19 | \( 1 - 361T \) |
| good | 7 | \( 1 - 42.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 210.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 674.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.99e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 717.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.46e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.49e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 7.34e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.30e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.51e5T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.03570721076268312731048370307, −9.046351471125225106057451848806, −8.234965177402689590495302785774, −7.20302340458588185397566217391, −6.67884645359082151334462989013, −5.25590790263357167506561845666, −4.54892940374298598621130430294, −2.98897877158509948128742994174, −1.80562036304419735133363815652, −0.38869611586015711434720167622,
0.38869611586015711434720167622, 1.80562036304419735133363815652, 2.98897877158509948128742994174, 4.54892940374298598621130430294, 5.25590790263357167506561845666, 6.67884645359082151334462989013, 7.20302340458588185397566217391, 8.234965177402689590495302785774, 9.046351471125225106057451848806, 10.03570721076268312731048370307
