L(s) = 1 | + (−1.02 − 0.425i)2-s + (0.165 + 0.165i)4-s + (−0.980 + 0.195i)5-s + (0.325 + 0.785i)8-s + (1.08 + 0.216i)10-s − 1.17i·16-s + (0.831 + 0.555i)17-s + (−0.707 + 1.70i)19-s + (−0.195 − 0.130i)20-s + (1.17 − 0.785i)23-s + (0.923 − 0.382i)25-s + (1.63 + 1.08i)31-s + (−0.176 + 0.425i)32-s + (−0.617 − 0.923i)34-s + (1.45 − 1.45i)38-s + ⋯ |
L(s) = 1 | + (−1.02 − 0.425i)2-s + (0.165 + 0.165i)4-s + (−0.980 + 0.195i)5-s + (0.325 + 0.785i)8-s + (1.08 + 0.216i)10-s − 1.17i·16-s + (0.831 + 0.555i)17-s + (−0.707 + 1.70i)19-s + (−0.195 − 0.130i)20-s + (1.17 − 0.785i)23-s + (0.923 − 0.382i)25-s + (1.63 + 1.08i)31-s + (−0.176 + 0.425i)32-s + (−0.617 − 0.923i)34-s + (1.45 − 1.45i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4466740220\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4466740220\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.980 - 0.195i)T \) |
| 17 | \( 1 + (-0.831 - 0.555i)T \) |
good | 2 | \( 1 + (1.02 + 0.425i)T + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 11 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 19 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \) |
| 37 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (-0.275 - 0.275i)T + iT^{2} \) |
| 53 | \( 1 + (-1.53 - 0.636i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
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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
−10.46018197848925540988559831921, −9.903191108964573339597112424900, −8.635290073365943857985782644452, −8.299200808184189317725413090119, −7.45216382100004641348242917006, −6.34785120656922232235895522749, −5.09471889277229155473594998411, −4.01647719670322119388976572929, −2.81836618054093303435389349861, −1.25131028205294972961245863721,
0.816155180189329016130218229162, 2.94710363602184265221011741971, 4.15002863137145274415409051657, 5.05507952016049189065111677994, 6.57299096343413816175048736469, 7.29939784073281190307430901899, 7.965007647169223652989169923320, 8.790876703277700439977616815864, 9.385960109557272621761689372229, 10.33882411427812998442888800156