| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 54·5-s + 36·6-s + 49·7-s − 64·8-s + 81·9-s + 216·10-s + 216·11-s − 144·12-s + 998·13-s − 196·14-s + 486·15-s + 256·16-s + 1.30e3·17-s − 324·18-s + 884·19-s − 864·20-s − 441·21-s − 864·22-s − 2.26e3·23-s + 576·24-s − 209·25-s − 3.99e3·26-s − 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.965·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.683·10-s + 0.538·11-s − 0.288·12-s + 1.63·13-s − 0.267·14-s + 0.557·15-s + 1/4·16-s + 1.09·17-s − 0.235·18-s + 0.561·19-s − 0.482·20-s − 0.218·21-s − 0.380·22-s − 0.893·23-s + 0.204·24-s − 0.0668·25-s − 1.15·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9035041258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9035041258\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 - 216 T + p^{5} T^{2} \) |
| 13 | \( 1 - 998 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1302 T + p^{5} T^{2} \) |
| 19 | \( 1 - 884 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2268 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1482 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8360 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4714 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9786 T + p^{5} T^{2} \) |
| 43 | \( 1 - 452 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 22200 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26790 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28092 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38866 T + p^{5} T^{2} \) |
| 67 | \( 1 - 23948 T + p^{5} T^{2} \) |
| 71 | \( 1 + 20628 T + p^{5} T^{2} \) |
| 73 | \( 1 - 290 T + p^{5} T^{2} \) |
| 79 | \( 1 + 99544 T + p^{5} T^{2} \) |
| 83 | \( 1 - 19308 T + p^{5} T^{2} \) |
| 89 | \( 1 - 36390 T + p^{5} T^{2} \) |
| 97 | \( 1 + 79078 T + p^{5} T^{2} \) |
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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
−15.47313583946692076142478319020, −13.91517915640706403416478175286, −12.09901367351987287270564556669, −11.45304040356271947065867993135, −10.25628162308257134178307377231, −8.622181411079888374791221407077, −7.51233742078421077201072399445, −5.94599798648278847718105300963, −3.85195247880333912368548057882, −1.01964620714483083984072115471,
1.01964620714483083984072115471, 3.85195247880333912368548057882, 5.94599798648278847718105300963, 7.51233742078421077201072399445, 8.622181411079888374791221407077, 10.25628162308257134178307377231, 11.45304040356271947065867993135, 12.09901367351987287270564556669, 13.91517915640706403416478175286, 15.47313583946692076142478319020
