L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 4·13-s − 2·14-s + 16-s − 6·17-s + 8·19-s + 20-s + 25-s + 4·26-s + 2·28-s + 31-s − 32-s + 6·34-s + 2·35-s − 4·37-s − 8·38-s − 40-s + 6·41-s + 8·43-s + 12·47-s − 3·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.83·19-s + 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.377·28-s + 0.179·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.657·37-s − 1.29·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.476909539\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.476909539\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.960813041623609009130793487464, −8.038084838934107691702825582988, −7.35265443309237049687362875942, −6.79822391154486169782162796968, −5.67746660566841356256437070474, −5.06269933860006516679132717000, −4.09880450289800587409359689727, −2.73830314365781363876465257615, −2.06128053911733423533178380014, −0.845490649179226708045903681820,
0.845490649179226708045903681820, 2.06128053911733423533178380014, 2.73830314365781363876465257615, 4.09880450289800587409359689727, 5.06269933860006516679132717000, 5.67746660566841356256437070474, 6.79822391154486169782162796968, 7.35265443309237049687362875942, 8.038084838934107691702825582988, 8.960813041623609009130793487464