| L(s) = 1 | + 0.877·2-s + 0.907·3-s − 1.22·4-s + 3.03·5-s + 0.796·6-s − 7-s − 2.83·8-s − 2.17·9-s + 2.66·10-s + 4.78·11-s − 1.11·12-s − 4.39·13-s − 0.877·14-s + 2.75·15-s − 0.0297·16-s + 17-s − 1.91·18-s − 2.64·19-s − 3.73·20-s − 0.907·21-s + 4.19·22-s − 8.45·23-s − 2.57·24-s + 4.23·25-s − 3.86·26-s − 4.69·27-s + 1.22·28-s + ⋯ |
| L(s) = 1 | + 0.620·2-s + 0.523·3-s − 0.614·4-s + 1.35·5-s + 0.325·6-s − 0.377·7-s − 1.00·8-s − 0.725·9-s + 0.843·10-s + 1.44·11-s − 0.322·12-s − 1.22·13-s − 0.234·14-s + 0.711·15-s − 0.00743·16-s + 0.242·17-s − 0.450·18-s − 0.606·19-s − 0.835·20-s − 0.198·21-s + 0.894·22-s − 1.76·23-s − 0.525·24-s + 0.846·25-s − 0.757·26-s − 0.904·27-s + 0.232·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.512251103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.512251103\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.877T + 2T^{2} \) |
| 3 | \( 1 - 0.907T + 3T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 11 | \( 1 - 4.78T + 11T^{2} \) |
| 13 | \( 1 + 4.39T + 13T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 8.45T + 23T^{2} \) |
| 29 | \( 1 - 7.04T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 9.66T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 - 9.28T + 59T^{2} \) |
| 61 | \( 1 - 6.66T + 61T^{2} \) |
| 67 | \( 1 - 5.28T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 1.94T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.75877616833833164512347688428, −12.70734124698541366659445798498, −11.79685478004627521729859500319, −9.891983029402487103373071879317, −9.451784805952369369026679333158, −8.377402703273289579004695071944, −6.45301199625848528619851078978, −5.59441068093551226192704092374, −4.08090296755249678835296246878, −2.49818869742691818855469877715,
2.49818869742691818855469877715, 4.08090296755249678835296246878, 5.59441068093551226192704092374, 6.45301199625848528619851078978, 8.377402703273289579004695071944, 9.451784805952369369026679333158, 9.891983029402487103373071879317, 11.79685478004627521729859500319, 12.70734124698541366659445798498, 13.75877616833833164512347688428
