L(s) = 1 | + 2-s − 1.57·3-s + 4-s + 5-s − 1.57·6-s − 4.21·7-s + 8-s − 0.503·9-s + 10-s − 5.96·11-s − 1.57·12-s − 3.37·13-s − 4.21·14-s − 1.57·15-s + 16-s + 0.836·17-s − 0.503·18-s + 3.55·19-s + 20-s + 6.66·21-s − 5.96·22-s + 4.00·23-s − 1.57·24-s + 25-s − 3.37·26-s + 5.53·27-s − 4.21·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.912·3-s + 0.5·4-s + 0.447·5-s − 0.645·6-s − 1.59·7-s + 0.353·8-s − 0.167·9-s + 0.316·10-s − 1.79·11-s − 0.456·12-s − 0.936·13-s − 1.12·14-s − 0.407·15-s + 0.250·16-s + 0.202·17-s − 0.118·18-s + 0.814·19-s + 0.223·20-s + 1.45·21-s − 1.27·22-s + 0.835·23-s − 0.322·24-s + 0.200·25-s − 0.662·26-s + 1.06·27-s − 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8346791779\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8346791779\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 + 5.96T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 - 0.836T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 0.622T + 53T^{2} \) |
| 59 | \( 1 + 8.49T + 59T^{2} \) |
| 61 | \( 1 - 8.67T + 61T^{2} \) |
| 67 | \( 1 + 2.96T + 67T^{2} \) |
| 71 | \( 1 + 1.88T + 71T^{2} \) |
| 73 | \( 1 + 2.28T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−7.70877940579285611963027803020, −7.20205482789109044954624206973, −6.47118557045396663322089274621, −5.78042731767050406877515999861, −5.26623082182179635748198461417, −4.86285243804498417537843780026, −3.38995580112928973858671369094, −3.01731539697227209103208083967, −2.10960166476769460013850289278, −0.41622988753789259631487005312,
0.41622988753789259631487005312, 2.10960166476769460013850289278, 3.01731539697227209103208083967, 3.38995580112928973858671369094, 4.86285243804498417537843780026, 5.26623082182179635748198461417, 5.78042731767050406877515999861, 6.47118557045396663322089274621, 7.20205482789109044954624206973, 7.70877940579285611963027803020