L(s) = 1 | + 2-s − i·3-s + 4-s + 2.93·5-s − i·6-s + (0.912 − 2.48i)7-s + 8-s − 9-s + 2.93·10-s + 4.68i·11-s − i·12-s − 0.902i·13-s + (0.912 − 2.48i)14-s − 2.93i·15-s + 16-s + 1.82·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s + 1.31·5-s − 0.408i·6-s + (0.344 − 0.938i)7-s + 0.353·8-s − 0.333·9-s + 0.927·10-s + 1.41i·11-s − 0.288i·12-s − 0.250i·13-s + (0.243 − 0.663i)14-s − 0.757i·15-s + 0.250·16-s + 0.442·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.97199 - 1.06103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.97199 - 1.06103i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.912 + 2.48i)T \) |
| 23 | \( 1 + (4.53 + 1.57i)T \) |
good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 - 4.68iT - 11T^{2} \) |
| 13 | \( 1 + 0.902iT - 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 - 5.58T + 19T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 3.09iT - 41T^{2} \) |
| 43 | \( 1 + 3.85iT - 43T^{2} \) |
| 47 | \( 1 - 0.610iT - 47T^{2} \) |
| 53 | \( 1 - 9.23iT - 53T^{2} \) |
| 59 | \( 1 + 12.3iT - 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 4.65iT - 67T^{2} \) |
| 71 | \( 1 - 8.37T + 71T^{2} \) |
| 73 | \( 1 - 5.51iT - 73T^{2} \) |
| 79 | \( 1 + 4.55iT - 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−9.934899771436775049356435505169, −9.400673243907619213995655281116, −7.82744677578507349572406367473, −7.35808264858749060779834139014, −6.43982563076722773290987416549, −5.57223822158347993668043162427, −4.80527727654261938704576134054, −3.61230670606523580383994197394, −2.23654992034401569302678734988, −1.44191531365675699045935681492,
1.70415801616692373578297891522, 2.81937761087033478582145778095, 3.75718651746706770295007472960, 5.19893834267086013701218943505, 5.67228189905466114787418391518, 6.14557086908977503999558890350, 7.57129545080418752603263770892, 8.622547038992784674132201544806, 9.406682889010764978679149950404, 10.01252629305003009486668059518