L(s) = 1 | + 1.41i·2-s − 2.23i·3-s + 2.23i·5-s + 3.16·6-s + (1.58 − 2.12i)7-s + 2.82i·8-s − 2.00·9-s − 3.16·10-s + (−3 − 1.41i)11-s − 6.32·13-s + (3 + 2.23i)14-s + 5.00·15-s − 4.00·16-s − 2.82i·18-s + 3.16·19-s + ⋯ |
L(s) = 1 | + 0.999i·2-s − 1.29i·3-s + 0.999i·5-s + 1.29·6-s + (0.597 − 0.801i)7-s + 0.999i·8-s − 0.666·9-s − 1.00·10-s + (−0.904 − 0.426i)11-s − 1.75·13-s + (0.801 + 0.597i)14-s + 1.29·15-s − 1.00·16-s − 0.666i·18-s + 0.725·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.978841 + 0.244609i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.978841 + 0.244609i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.58 + 2.12i)T \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
good | 2 | \( 1 - 1.41iT - 2T^{2} \) |
| 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 5 | \( 1 - 2.23iT - 5T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.70iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 4.47iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2.23iT - 59T^{2} \) |
| 61 | \( 1 - 3.16T + 61T^{2} \) |
| 67 | \( 1 - 11T + 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 3.16T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 2.23iT - 89T^{2} \) |
| 97 | \( 1 + 6.70iT - 97T^{2} \) |
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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
−14.30907802043378142686202147827, −14.00239147289052012476694589578, −12.53509818569497510373937051691, −11.36932173789363899884118231400, −10.28226613420318595849526213367, −8.072187683008054892937878537238, −7.34063380060344316724817172191, −6.82062635626822745794787763240, −5.28719110765115007953962567184, −2.47464235229053040124719550329,
2.48357509766793217070371542497, 4.41761480213862959022213035276, 5.29394234540745949716918476449, 7.76468001337108502788456218067, 9.395875319706378008010643248436, 9.864859786780334358578785974913, 11.09194271487551417841285780592, 12.13155401606035181292274243983, 12.83183722072605330609778214000, 14.62952926544691300205816133475