L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)8-s + (1.17 + 0.984i)11-s + (0.766 + 0.642i)16-s + (0.939 + 1.62i)17-s + (−0.173 + 0.300i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)25-s + (0.766 − 0.642i)32-s + (1.76 − 0.642i)34-s + (0.266 + 0.223i)38-s + (0.0603 + 0.342i)41-s + (−1.43 − 1.20i)43-s + (−0.766 − 1.32i)44-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.5 + 0.866i)8-s + (1.17 + 0.984i)11-s + (0.766 + 0.642i)16-s + (0.939 + 1.62i)17-s + (−0.173 + 0.300i)19-s + (1.17 − 0.984i)22-s + (0.173 − 0.984i)25-s + (0.766 − 0.642i)32-s + (1.76 − 0.642i)34-s + (0.266 + 0.223i)38-s + (0.0603 + 0.342i)41-s + (−1.43 − 1.20i)43-s + (−0.766 − 1.32i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.204916089\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204916089\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
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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
−9.471067256635339345762760029218, −8.631192401909108119143266667319, −8.004269583820269675503443290453, −6.79818309452422434409409088118, −6.01409961405991888550015379226, −5.05186480967533061509192256783, −4.07270392229916274098671787366, −3.58947708986422348532754412386, −2.20895382781020691492426975177, −1.35981202647309825896168753469,
1.02700077572391596750237886066, 3.01224616975296691773175399169, 3.73347602510536179227797238339, 4.81770258966898918946966693137, 5.52325684391100631712637779192, 6.37563602507622255187596912409, 7.05546286392487499445781236814, 7.80017948548131073751146776421, 8.667706698449084858542184609098, 9.299595254252309832904547843738