L(s) = 1 | + (0.900 + 1.56i)2-s + (0.222 − 0.385i)3-s + (−1.12 + 1.94i)4-s + (−0.623 − 1.07i)5-s + 0.801·6-s − 2.24·8-s + (0.400 + 0.694i)9-s + (1.12 − 1.94i)10-s + (0.499 + 0.866i)12-s − 0.554·15-s + (−0.900 − 1.56i)16-s + (−0.722 + 1.25i)18-s + (0.900 + 1.56i)19-s + 2.80·20-s + (−0.500 + 0.866i)24-s + (−0.277 + 0.480i)25-s + ⋯ |
L(s) = 1 | + (0.900 + 1.56i)2-s + (0.222 − 0.385i)3-s + (−1.12 + 1.94i)4-s + (−0.623 − 1.07i)5-s + 0.801·6-s − 2.24·8-s + (0.400 + 0.694i)9-s + (1.12 − 1.94i)10-s + (0.499 + 0.866i)12-s − 0.554·15-s + (−0.900 − 1.56i)16-s + (−0.722 + 1.25i)18-s + (0.900 + 1.56i)19-s + 2.80·20-s + (−0.500 + 0.866i)24-s + (−0.277 + 0.480i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3479 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.863415741\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863415741\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + 0.445T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.80T + T^{2} \) |
| 89 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.441034267472186927665451690212, −8.078942307271658332946867142315, −7.56559821927742076449693893252, −6.89800560543152964627633158942, −5.89635184057211227287610790269, −5.35443709383948362727422562914, −4.51973035955392498793463074680, −4.07739474649652123509236687199, −3.01985465048293395942077836386, −1.42180460640654870876114204339,
0.897539491788389904159860570740, 2.37721643520263157803459343976, 2.99641673417018749870916723505, 3.69986542689964664009349716993, 4.25067760423411041997258828861, 5.10853056465199984290097406416, 6.04465302146776748786762552219, 6.98580662566909687561380056011, 7.61973360320699801492304775193, 9.069431769878939429752676025592