L(s) = 1 | + (0.800 + 0.800i)2-s + (−1.09 − 1.34i)3-s − 0.718i·4-s + (2.10 − 0.754i)5-s + (0.199 − 1.95i)6-s + (−0.707 + 0.707i)7-s + (2.17 − 2.17i)8-s + (−0.606 + 2.93i)9-s + (2.28 + 1.08i)10-s + 5.20i·11-s + (−0.964 + 0.785i)12-s + (−3.24 − 3.24i)13-s − 1.13·14-s + (−3.31 − 2.00i)15-s + 2.04·16-s + (0.844 + 0.844i)17-s + ⋯ |
L(s) = 1 | + (0.566 + 0.566i)2-s + (−0.631 − 0.775i)3-s − 0.359i·4-s + (0.941 − 0.337i)5-s + (0.0813 − 0.796i)6-s + (−0.267 + 0.267i)7-s + (0.769 − 0.769i)8-s + (−0.202 + 0.979i)9-s + (0.723 + 0.341i)10-s + 1.56i·11-s + (−0.278 + 0.226i)12-s + (−0.900 − 0.900i)13-s − 0.302·14-s + (−0.856 − 0.516i)15-s + 0.511·16-s + (0.204 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20946 - 0.189608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20946 - 0.189608i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.09 + 1.34i)T \) |
| 5 | \( 1 + (-2.10 + 0.754i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (-0.800 - 0.800i)T + 2iT^{2} \) |
| 11 | \( 1 - 5.20iT - 11T^{2} \) |
| 13 | \( 1 + (3.24 + 3.24i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.844 - 0.844i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.32iT - 19T^{2} \) |
| 23 | \( 1 + (5.62 - 5.62i)T - 23iT^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 + (1.71 - 1.71i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 + (0.281 + 0.281i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.39 + 3.39i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.51 + 3.51i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 + (-7.92 + 7.92i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.06iT - 71T^{2} \) |
| 73 | \( 1 + (1.33 + 1.33i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 + (5.46 - 5.46i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 + (3.06 - 3.06i)T - 97iT^{2} \) |
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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
−13.64694242910135106182736411076, −12.78074681911096866117436862216, −12.12393888649730160483569988036, −10.29175454439059112760219584415, −9.756157406457628016915272588255, −7.75699758381480193502843083159, −6.69171445998239467331410681333, −5.66093781661780693293120598090, −4.87718749030754123064819634956, −1.88209981289986690488224305437,
2.81380028136633989558056339741, 4.20978539455641778155056428383, 5.53571340099068463547345076261, 6.74497994914376859413233139139, 8.633888192996722617610450202862, 9.886907270414680614812934020707, 10.80475995513307109176466318938, 11.66010697887623285967427188563, 12.66752643302369429647103741802, 13.93692566294017443107492849548