L(s) = 1 | + 0.578i·2-s − 0.551i·3-s + 1.66·4-s + (1.93 + 1.12i)5-s + 0.319·6-s − 4.66i·7-s + 2.11i·8-s + 2.69·9-s + (−0.650 + 1.11i)10-s − 1.22·11-s − 0.919i·12-s − 5.34i·13-s + 2.69·14-s + (0.620 − 1.06i)15-s + 2.10·16-s + 1.41i·17-s + ⋯ |
L(s) = 1 | + 0.408i·2-s − 0.318i·3-s + 0.832·4-s + (0.864 + 0.503i)5-s + 0.130·6-s − 1.76i·7-s + 0.749i·8-s + 0.898·9-s + (−0.205 + 0.353i)10-s − 0.369·11-s − 0.265i·12-s − 1.48i·13-s + 0.720·14-s + (0.160 − 0.275i)15-s + 0.526·16-s + 0.342i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.680911496\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.680911496\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.93 - 1.12i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 0.578iT - 2T^{2} \) |
| 3 | \( 1 + 0.551iT - 3T^{2} \) |
| 7 | \( 1 + 4.66iT - 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 5.34iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 23 | \( 1 + 1.95iT - 23T^{2} \) |
| 29 | \( 1 + 7.32T + 29T^{2} \) |
| 31 | \( 1 + 1.83T + 31T^{2} \) |
| 37 | \( 1 + 5.59iT - 37T^{2} \) |
| 41 | \( 1 - 8.29T + 41T^{2} \) |
| 43 | \( 1 + 8.30iT - 43T^{2} \) |
| 47 | \( 1 - 4.10iT - 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 - 4.65iT - 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 4.99iT - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 1.02iT - 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 - 4.72iT - 97T^{2} \) |
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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
−9.338412571220235379766878084315, −7.901910321814272906140718362565, −7.46968643963490455200983434927, −7.02557776858123744448296384287, −6.11216154020907628062223305525, −5.45957704352634275938225361175, −4.18930702471175161017433046986, −3.17834295627870042722642794698, −2.08101576329719828411181012256, −1.00451971208845258222304363303,
1.68544183079392578049080933889, 2.09803389982137216318938034902, 3.17235111132651730890723961748, 4.46171406293461845413773368914, 5.32939431124837865425872189426, 6.09269166665759305315765342598, 6.77878056020369558099347231692, 7.78931703186635733486564000525, 8.906493766463536148101191332718, 9.482295294891164103483250962712