L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 1.22i)5-s + (−0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + (−1 + 0.999i)10-s + (0.866 + 0.5i)13-s + (−0.258 − 0.965i)14-s + (0.500 + 0.866i)16-s + 1.41i·17-s + i·19-s + (−1.22 + 0.707i)20-s + (−1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (0.707 + 0.707i)26-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 1.22i)5-s + (−0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + (−1 + 0.999i)10-s + (0.866 + 0.5i)13-s + (−0.258 − 0.965i)14-s + (0.500 + 0.866i)16-s + 1.41i·17-s + i·19-s + (−1.22 + 0.707i)20-s + (−1.22 − 0.707i)23-s + (−0.499 − 0.866i)25-s + (0.707 + 0.707i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.791516129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.791516129\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.879927836108669322274161049743, −8.316342984026286049345620146311, −7.86802138551586987442565991298, −7.00961512698481437764824393019, −6.32729224529124223321803877214, −5.90954167212259057848357839537, −4.18776030151012489437413780541, −3.94916829315224528102807916422, −3.16899701042474434620135318813, −1.92480651337774270848425548859,
1.00783614577151169499100071847, 2.49221584353409145990665859967, 3.39402786809487850388162923236, 4.33137991996801459195225426831, 5.12452840590224191445349673569, 5.70583645759446406069060542519, 6.65445065339060837349879176631, 7.58469156620135781392897183494, 8.477591926304119729735823806122, 9.160354188644758802538982061915