| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 14.2·5-s + 36·6-s + 60.7·7-s − 64·8-s + 81·9-s + 56.8·10-s + 675.·11-s − 144·12-s + 758.·13-s − 242.·14-s + 127.·15-s + 256·16-s − 102.·17-s − 324·18-s − 984.·19-s − 227.·20-s − 546.·21-s − 2.70e3·22-s + 77.2·23-s + 576·24-s − 2.92e3·25-s − 3.03e3·26-s − 729·27-s + 971.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.254·5-s + 0.408·6-s + 0.468·7-s − 0.353·8-s + 0.333·9-s + 0.179·10-s + 1.68·11-s − 0.288·12-s + 1.24·13-s − 0.331·14-s + 0.146·15-s + 0.250·16-s − 0.0859·17-s − 0.235·18-s − 0.625·19-s − 0.127·20-s − 0.270·21-s − 1.19·22-s + 0.0304·23-s + 0.204·24-s − 0.935·25-s − 0.880·26-s − 0.192·27-s + 0.234·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.436853895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436853895\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 + 14.2T + 3.12e3T^{2} \) |
| 7 | \( 1 - 60.7T + 1.68e4T^{2} \) |
| 11 | \( 1 - 675.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 758.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 102.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 984.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 77.2T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.95e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.07e3T + 4.18e8T^{2} \) |
| 61 | \( 1 - 7.97e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.33e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.68e5T + 8.58e9T^{2} \) |
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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
−10.68724401086495467392744272470, −9.742050021835961967882413703431, −8.710042502096132177684414737024, −8.031850255476080359958980288316, −6.62138620899812063249446731046, −6.22174390890592704793415430419, −4.63684498954185625589933704043, −3.57490857694018078936123309550, −1.72450746605498179717825897961, −0.799395774092365461821550347823,
0.799395774092365461821550347823, 1.72450746605498179717825897961, 3.57490857694018078936123309550, 4.63684498954185625589933704043, 6.22174390890592704793415430419, 6.62138620899812063249446731046, 8.031850255476080359958980288316, 8.710042502096132177684414737024, 9.742050021835961967882413703431, 10.68724401086495467392744272470
