L(s) = 1 | + (−0.690 − 1.87i)2-s + (−3.04 + 2.59i)4-s + 8.91·5-s + 7.74i·7-s + (6.96 + 3.93i)8-s + (−6.15 − 16.7i)10-s − 16.0i·11-s − 6.44·13-s + (14.5 − 5.34i)14-s + (2.57 − 15.7i)16-s − 1.80·17-s − 4.35i·19-s + (−27.1 + 23.0i)20-s + (−30.1 + 11.0i)22-s + 9.27i·23-s + ⋯ |
L(s) = 1 | + (−0.345 − 0.938i)2-s + (−0.761 + 0.647i)4-s + 1.78·5-s + 1.10i·7-s + (0.870 + 0.491i)8-s + (−0.615 − 1.67i)10-s − 1.46i·11-s − 0.495·13-s + (1.03 − 0.381i)14-s + (0.160 − 0.987i)16-s − 0.106·17-s − 0.229i·19-s + (−1.35 + 1.15i)20-s + (−1.37 + 0.504i)22-s + 0.403i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.000925284\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.000925284\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.690 + 1.87i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 - 8.91T + 25T^{2} \) |
| 7 | \( 1 - 7.74iT - 49T^{2} \) |
| 11 | \( 1 + 16.0iT - 121T^{2} \) |
| 13 | \( 1 + 6.44T + 169T^{2} \) |
| 17 | \( 1 + 1.80T + 289T^{2} \) |
| 23 | \( 1 - 9.27iT - 529T^{2} \) |
| 29 | \( 1 - 40.7T + 841T^{2} \) |
| 31 | \( 1 + 25.0iT - 961T^{2} \) |
| 37 | \( 1 - 71.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 49.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 82.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 35.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 80.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 51.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.89iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 117.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.35iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 91.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 87.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 107.T + 9.40e3T^{2} \) |
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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
−10.00501108626906650529115118608, −9.399223609998446876754405880381, −8.840415070499268634211522381882, −7.910827286469563528549089152801, −6.24086881457066711971416491556, −5.71460179317380805672676505921, −4.66194420163151852928559022363, −2.89835971031804581569975331657, −2.41765209490049239335735684426, −1.06725797421089297290736818578,
1.09404611084877322679755506030, 2.30902732157313360479967882891, 4.34176392045052138261453845256, 5.04999677775605222879232530329, 6.12585156407232361937644751937, 6.86575299985169549402621084402, 7.52932441100409885418045645838, 8.766582819660377277425341509597, 9.659156113014768983461827038523, 10.12645047398814916642256695773