| L(s) = 1 | − 1.71·2-s − 3.02·3-s + 0.944·4-s + 5-s + 5.18·6-s − 0.410·7-s + 1.81·8-s + 6.12·9-s − 1.71·10-s − 6.26·11-s − 2.85·12-s − 4.80·13-s + 0.704·14-s − 3.02·15-s − 4.99·16-s − 17-s − 10.5·18-s + 0.789·19-s + 0.944·20-s + 1.24·21-s + 10.7·22-s + 5.84·23-s − 5.47·24-s + 25-s + 8.24·26-s − 9.44·27-s − 0.387·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 1.74·3-s + 0.472·4-s + 0.447·5-s + 2.11·6-s − 0.155·7-s + 0.640·8-s + 2.04·9-s − 0.542·10-s − 1.88·11-s − 0.823·12-s − 1.33·13-s + 0.188·14-s − 0.780·15-s − 1.24·16-s − 0.242·17-s − 2.47·18-s + 0.181·19-s + 0.211·20-s + 0.270·21-s + 2.29·22-s + 1.21·23-s − 1.11·24-s + 0.200·25-s + 1.61·26-s − 1.81·27-s − 0.0733·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 3 | \( 1 + 3.02T + 3T^{2} \) |
| 7 | \( 1 + 0.410T + 7T^{2} \) |
| 11 | \( 1 + 6.26T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 19 | \( 1 - 0.789T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 3.69T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 7.78T + 47T^{2} \) |
| 53 | \( 1 + 4.68T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 73 | \( 1 + 1.61T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 0.415T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
| 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
−7.59923644005214607828298348638, −7.13553154501581685352364422209, −6.43696190561797071782240793626, −5.50521210765264774165001514449, −4.95813732499169391215754156283, −4.60164936369748188246325868446, −2.89401998010317424324304826756, −1.95106254655313501850634908193, −0.77330210068529480575344098956, 0,
0.77330210068529480575344098956, 1.95106254655313501850634908193, 2.89401998010317424324304826756, 4.60164936369748188246325868446, 4.95813732499169391215754156283, 5.50521210765264774165001514449, 6.43696190561797071782240793626, 7.13553154501581685352364422209, 7.59923644005214607828298348638
