L(s) = 1 | − 1.72i·2-s − i·3-s − 0.979·4-s − 1.65i·5-s − 1.72·6-s − 1.76i·8-s − 9-s − 2.85·10-s + (−2.71 − 1.89i)11-s + 0.979i·12-s − 2.97·13-s − 1.65·15-s − 4.99·16-s + 1.49·17-s + 1.72i·18-s + 5.51·19-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.577i·3-s − 0.489·4-s − 0.740i·5-s − 0.704·6-s − 0.622i·8-s − 0.333·9-s − 0.903·10-s + (−0.820 − 0.572i)11-s + 0.282i·12-s − 0.825·13-s − 0.427·15-s − 1.24·16-s + 0.362·17-s + 0.406i·18-s + 1.26·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.000882086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.000882086\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.71 + 1.89i)T \) |
good | 2 | \( 1 + 1.72iT - 2T^{2} \) |
| 5 | \( 1 + 1.65iT - 5T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 + 8.62T + 23T^{2} \) |
| 29 | \( 1 + 8.02iT - 29T^{2} \) |
| 31 | \( 1 - 8.86iT - 31T^{2} \) |
| 37 | \( 1 - 8.22T + 37T^{2} \) |
| 41 | \( 1 + 8.12T + 41T^{2} \) |
| 43 | \( 1 + 5.03iT - 43T^{2} \) |
| 47 | \( 1 - 7.36iT - 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 + 4.96iT - 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 + 9.85T + 73T^{2} \) |
| 79 | \( 1 + 3.93iT - 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.16iT - 89T^{2} \) |
| 97 | \( 1 + 3.35iT - 97T^{2} \) |
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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
−8.966693061025021779232932040227, −8.035051774166409260281803697237, −7.44421327136393799873068830953, −6.30861418482113731150800387544, −5.38834120258502250349851857946, −4.49440450616669571854598275895, −3.31855437533739201863901898325, −2.52875011474547016404067497430, −1.47265556505313521120961545079, −0.36878931645768760885798676521,
2.23840531825047729130719965401, 3.15809471455253266112969473636, 4.45987168535509088940144446709, 5.25941385452522960156936573284, 5.92095731573434145267175100008, 6.86650081590153959863920590277, 7.59406918011320746178402667353, 7.998257764178512035619011054222, 9.138438484630100148252739016615, 9.973126310611043370453921325861