L(s) = 1 | − 2-s + 4.37·3-s − 3·4-s + 3.17i·5-s − 4.37·6-s + (−0.818 − 6.95i)7-s + 7·8-s + 10.1·9-s − 3.17i·10-s − 19.4i·11-s − 13.1·12-s − 4.37·13-s + (0.818 + 6.95i)14-s + 13.9i·15-s + 5·16-s + 25.3·17-s + ⋯ |
L(s) = 1 | − 0.5·2-s + 1.45·3-s − 0.750·4-s + 0.635i·5-s − 0.729·6-s + (−0.116 − 0.993i)7-s + 0.875·8-s + 1.12·9-s − 0.317i·10-s − 1.76i·11-s − 1.09·12-s − 0.336·13-s + (0.0584 + 0.496i)14-s + 0.926i·15-s + 0.312·16-s + 1.49·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.801 + 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.62644 - 0.539830i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62644 - 0.539830i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.818 + 6.95i)T \) |
| 41 | \( 1 + (-28.1 + 29.7i)T \) |
good | 2 | \( 1 + T + 4T^{2} \) |
| 3 | \( 1 - 4.37T + 9T^{2} \) |
| 5 | \( 1 - 3.17iT - 25T^{2} \) |
| 11 | \( 1 + 19.4iT - 121T^{2} \) |
| 13 | \( 1 + 4.37T + 169T^{2} \) |
| 17 | \( 1 - 25.3T + 289T^{2} \) |
| 19 | \( 1 - 23.3T + 361T^{2} \) |
| 23 | \( 1 - 0.747T + 529T^{2} \) |
| 29 | \( 1 - 11.3iT - 841T^{2} \) |
| 31 | \( 1 - 17.7iT - 961T^{2} \) |
| 37 | \( 1 + 37.1T + 1.36e3T^{2} \) |
| 43 | \( 1 - 35.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 69.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 20.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 75.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 61.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 19.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 46.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 165. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 70.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 100.T + 9.40e3T^{2} \) |
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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
−11.16333981883852669337574056046, −10.23495288367159389077867842548, −9.543643276033569719341771829534, −8.565963773852478659080298043686, −7.892729954000618277170305767542, −7.08223139297054363711633325270, −5.34184589417762065170149194016, −3.62931433385231657935234952004, −3.18679382191528149114816429107, −1.00842308874681905703566984295,
1.58259101447336891926685299606, 2.99240006365398813832655668845, 4.41595303150734273438275784563, 5.39108232544212100472307221215, 7.45450464740132194865138167698, 7.969308957173813728076218314628, 9.045420515580751340221950576199, 9.474720840246410246181368599211, 10.08627836156027687551490863214, 12.07855682239202690720881174819