L(s) = 1 | + 10.2i·2-s + (12.6 + 23.8i)3-s − 41.9·4-s + 93.4i·5-s + (−245. + 129. i)6-s + 330.·7-s + 226. i·8-s + (−410. + 602. i)9-s − 961.·10-s + 306. i·11-s + (−529. − 1.00e3i)12-s − 1.04e3·13-s + 3.40e3i·14-s + (−2.23e3 + 1.17e3i)15-s − 5.02e3·16-s − 1.08e3i·17-s + ⋯ |
L(s) = 1 | + 1.28i·2-s + (0.467 + 0.884i)3-s − 0.655·4-s + 0.747i·5-s + (−1.13 + 0.601i)6-s + 0.963·7-s + 0.442i·8-s + (−0.563 + 0.826i)9-s − 0.961·10-s + 0.230i·11-s + (−0.306 − 0.579i)12-s − 0.476·13-s + 1.24i·14-s + (−0.660 + 0.349i)15-s − 1.22·16-s − 0.220i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.541154 - 2.18189i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541154 - 2.18189i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-12.6 - 23.8i)T \) |
| 23 | \( 1 + 2.53e3iT \) |
good | 2 | \( 1 - 10.2iT - 64T^{2} \) |
| 5 | \( 1 - 93.4iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 330.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 306. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 1.04e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.08e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.64e3T + 4.70e7T^{2} \) |
| 29 | \( 1 + 4.45e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 8.54e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.22e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.78e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.30e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.20e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.62e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.98e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 4.30e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 1.11e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.67e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 9.28e3T + 1.51e11T^{2} \) |
| 79 | \( 1 - 2.75e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 1.05e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.07e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.03e5T + 8.32e11T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.36094335847459365433519532290, −13.91805913454739879296668961977, −11.69246192744957706832966063335, −10.66979478985949620289316075226, −9.349822387001590574634873187667, −8.068936914293641736726131485405, −7.24978271735014921289646601126, −5.63386116978849240566961893819, −4.50427561873362607316566730423, −2.57384546600113723250233607837,
0.875832284961433303753491735626, 1.82808400995299756120282904057, 3.31961341076071082277533803172, 5.08537242592053759359877623456, 7.09335314807812356317401045190, 8.402272655719158907034331908328, 9.427805230300574481035407094045, 10.92307308707975247522410587664, 11.95114112900492107273748619124, 12.58593661692838208021449374658