L(s) = 1 | − 17.7i·2-s − 27i·3-s − 188.·4-s + (57.4 − 273. i)5-s − 480.·6-s + 1.23e3i·7-s + 1.06e3i·8-s − 729·9-s + (−4.86e3 − 1.02e3i)10-s + 1.21e3·11-s + 5.08e3i·12-s − 1.35e4i·13-s + 2.19e4·14-s + (−7.38e3 − 1.55e3i)15-s − 5.06e3·16-s − 1.20e4i·17-s + ⋯ |
L(s) = 1 | − 1.57i·2-s − 0.577i·3-s − 1.47·4-s + (0.205 − 0.978i)5-s − 0.907·6-s + 1.35i·7-s + 0.738i·8-s − 0.333·9-s + (−1.53 − 0.323i)10-s + 0.275·11-s + 0.848i·12-s − 1.70i·13-s + 2.13·14-s + (−0.565 − 0.118i)15-s − 0.308·16-s − 0.597i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.140204 + 1.34857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.140204 + 1.34857i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27iT \) |
| 5 | \( 1 + (-57.4 + 273. i)T \) |
good | 2 | \( 1 + 17.7iT - 128T^{2} \) |
| 7 | \( 1 - 1.23e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.35e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 3.28e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.42e3iT - 3.40e9T^{2} \) |
| 29 | \( 1 - 7.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.13e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.18e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 6.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.91e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 3.05e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.44e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.00e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.79e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 3.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.67e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 3.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.77e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 4.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.60e6iT - 8.07e13T^{2} \) |
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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
−17.67265084260284939308125492369, −15.64917163460973452792046371781, −13.51751364019418356548337161476, −12.45385650518820746335479342197, −11.73783287282642311097357926754, −9.796681432451493697699143613069, −8.482496132848032876617861973803, −5.36399523029994110584349003619, −2.74274877496899097665027724920, −0.953046337948809790252220674156,
4.22220423891170523818833110192, 6.36907659384206059251473587088, 7.48959171795985101714205964839, 9.479392567865659343612725815761, 11.06725465341160243387854588104, 13.98603980755424067437475965143, 14.33838198740892458262410255621, 15.89279851394759522995092248160, 16.83798856339034651587528160613, 17.85095227886717253230004196618