L(s) = 1 | − 0.879·2-s − 3-s − 1.22·4-s + 3.40·5-s + 0.879·6-s + 5.00·7-s + 2.83·8-s + 9-s − 2.99·10-s − 11-s + 1.22·12-s − 4.39·14-s − 3.40·15-s − 0.0436·16-s + 5.17·17-s − 0.879·18-s + 1.09·19-s − 4.18·20-s − 5.00·21-s + 0.879·22-s − 2.47·23-s − 2.83·24-s + 6.62·25-s − 27-s − 6.13·28-s + 4.36·29-s + 2.99·30-s + ⋯ |
L(s) = 1 | − 0.621·2-s − 0.577·3-s − 0.613·4-s + 1.52·5-s + 0.359·6-s + 1.89·7-s + 1.00·8-s + 0.333·9-s − 0.948·10-s − 0.301·11-s + 0.353·12-s − 1.17·14-s − 0.880·15-s − 0.0109·16-s + 1.25·17-s − 0.207·18-s + 0.251·19-s − 0.934·20-s − 1.09·21-s + 0.187·22-s − 0.515·23-s − 0.579·24-s + 1.32·25-s − 0.192·27-s − 1.15·28-s + 0.811·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.922246385\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.922246385\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 0.879T + 2T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 - 5.00T + 7T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 4.36T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 + 1.87T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 8.84T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 + 3.18T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.171291102052226251107686359446, −7.66024345972170606943243846475, −6.79654544003449638682084691259, −5.55866596867430686486278183216, −5.46239216920621760536498851128, −4.78580013444194292991508217431, −3.89770321348559856109579975520, −2.37167722827121749121614519290, −1.56564837450106310122058445657, −0.961159121507201813993646431720,
0.961159121507201813993646431720, 1.56564837450106310122058445657, 2.37167722827121749121614519290, 3.89770321348559856109579975520, 4.78580013444194292991508217431, 5.46239216920621760536498851128, 5.55866596867430686486278183216, 6.79654544003449638682084691259, 7.66024345972170606943243846475, 8.171291102052226251107686359446