L(s) = 1 | + 2-s − 1.49·3-s + 4-s − 5-s − 1.49·6-s − 0.532·7-s + 8-s − 0.769·9-s − 10-s − 11-s − 1.49·12-s − 2.49·13-s − 0.532·14-s + 1.49·15-s + 16-s + 4.31·17-s − 0.769·18-s − 7.60·19-s − 20-s + 0.794·21-s − 22-s − 4.95·23-s − 1.49·24-s + 25-s − 2.49·26-s + 5.62·27-s − 0.532·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.862·3-s + 0.5·4-s − 0.447·5-s − 0.609·6-s − 0.201·7-s + 0.353·8-s − 0.256·9-s − 0.316·10-s − 0.301·11-s − 0.431·12-s − 0.692·13-s − 0.142·14-s + 0.385·15-s + 0.250·16-s + 1.04·17-s − 0.181·18-s − 1.74·19-s − 0.223·20-s + 0.173·21-s − 0.213·22-s − 1.03·23-s − 0.304·24-s + 0.200·25-s − 0.489·26-s + 1.08·27-s − 0.100·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.133260944\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.133260944\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 7 | \( 1 + 0.532T + 7T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 + 7.60T + 19T^{2} \) |
| 23 | \( 1 + 4.95T + 23T^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 + 3.47T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 0.259T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.61T + 67T^{2} \) |
| 71 | \( 1 - 7.33T + 71T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 5.76T + 89T^{2} \) |
| 97 | \( 1 - 8.19T + 97T^{2} \) |
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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
−7.78636278334927844251941895542, −6.89597519213921475265659327338, −6.33296518049021988431683617509, −5.70337274258973540783478002781, −5.07667750172746639540386445067, −4.40455489877750407533842812523, −3.63989939908165236648945654035, −2.78818532831393805410819949275, −1.90786888967037074882733269376, −0.47014019650910528806089726569,
0.47014019650910528806089726569, 1.90786888967037074882733269376, 2.78818532831393805410819949275, 3.63989939908165236648945654035, 4.40455489877750407533842812523, 5.07667750172746639540386445067, 5.70337274258973540783478002781, 6.33296518049021988431683617509, 6.89597519213921475265659327338, 7.78636278334927844251941895542