L(s) = 1 | + i·2-s + (1.23 + 1.21i)3-s − 4-s − 0.666·5-s + (−1.21 + 1.23i)6-s + i·7-s − i·8-s + (0.0269 + 2.99i)9-s − 0.666i·10-s − 5.34·11-s + (−1.23 − 1.21i)12-s + 5.36·13-s − 14-s + (−0.820 − 0.812i)15-s + 16-s − 0.110·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.710 + 0.703i)3-s − 0.5·4-s − 0.298·5-s + (−0.497 + 0.502i)6-s + 0.377i·7-s − 0.353i·8-s + (0.00899 + 0.999i)9-s − 0.210i·10-s − 1.61·11-s + (−0.355 − 0.351i)12-s + 1.48·13-s − 0.267·14-s + (−0.211 − 0.209i)15-s + 0.250·16-s − 0.0267·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0528748 - 1.27026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0528748 - 1.27026i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-1.23 - 1.21i)T \) |
| 7 | \( 1 - iT \) |
| 23 | \( 1 + (3.11 - 3.64i)T \) |
good | 5 | \( 1 + 0.666T + 5T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 - 5.36T + 13T^{2} \) |
| 17 | \( 1 + 0.110T + 17T^{2} \) |
| 19 | \( 1 - 7.89iT - 19T^{2} \) |
| 29 | \( 1 + 5.21iT - 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 + 4.98iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.96iT - 43T^{2} \) |
| 47 | \( 1 + 4.18iT - 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 - 2.11iT - 59T^{2} \) |
| 61 | \( 1 - 3.58iT - 61T^{2} \) |
| 67 | \( 1 - 8.90iT - 67T^{2} \) |
| 71 | \( 1 - 6.81iT - 71T^{2} \) |
| 73 | \( 1 - 9.22T + 73T^{2} \) |
| 79 | \( 1 - 5.18iT - 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 + 1.57iT - 97T^{2} \) |
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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
−10.22830485080677012551901653022, −9.582034272952610081536645019114, −8.452900554342373399117961910356, −8.122366209282428167550624850931, −7.41117005824291425427544642118, −5.79007657780332110678715871852, −5.53547034780743487709418121842, −4.09152129243907149051837208944, −3.51939273852841819589453060530, −2.08039714920297047191528735285,
0.52327513940215039315776478819, 2.00365794450986192608402019362, 3.01214508914486771642507276481, 3.87103284282545239153913297915, 5.04901525475137394919293870491, 6.24436858389096126505228321724, 7.28151196105638241624239175376, 8.052302381418632466313304954811, 8.705418842001629115981012693964, 9.509138278105462014948772652840