L(s) = 1 | + (−1.03 − 0.965i)2-s + (0.133 + 1.99i)4-s − 2.73i·5-s + (1.78 − 2.19i)8-s + (−2.63 + 2.82i)10-s + 5.64·11-s + 1.41·13-s + (−3.96 + 0.534i)16-s + 6.19i·17-s + 4.13i·19-s + (5.45 − 0.366i)20-s + (−5.83 − 5.45i)22-s + 5.64·23-s − 2.46·25-s + (−1.46 − 1.36i)26-s + ⋯ |
L(s) = 1 | + (−0.730 − 0.683i)2-s + (0.0669 + 0.997i)4-s − 1.22i·5-s + (0.632 − 0.774i)8-s + (−0.834 + 0.892i)10-s + 1.70·11-s + 0.392·13-s + (−0.991 + 0.133i)16-s + 1.50i·17-s + 0.947i·19-s + (1.21 − 0.0818i)20-s + (−1.24 − 1.16i)22-s + 1.17·23-s − 0.492·25-s + (−0.286 − 0.267i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.345385261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345385261\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.03 + 0.965i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 6.19iT - 17T^{2} \) |
| 19 | \( 1 - 4.13iT - 19T^{2} \) |
| 23 | \( 1 - 5.64T + 23T^{2} \) |
| 29 | \( 1 - 0.378iT - 29T^{2} \) |
| 31 | \( 1 - 7.15iT - 31T^{2} \) |
| 37 | \( 1 + 3.46T + 37T^{2} \) |
| 41 | \( 1 - 5.26iT - 41T^{2} \) |
| 43 | \( 1 - 5.84iT - 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 0.656iT - 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 + 7.98iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 13.8iT - 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.26iT - 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.167830690157723946231317095781, −8.567161747446572168667159033366, −8.129862682349496724452357884707, −6.90033604739247849695066332840, −6.17606353526347572174734219284, −4.90563235222907669637809625537, −4.02775666707237829878608608505, −3.34680426187155063433943586841, −1.59528905667869107886989077524, −1.19239835061893746890998074534,
0.804631926316138298542881284945, 2.24958134113888676503354150447, 3.34715352987697803388620731280, 4.53069638292010809357888116348, 5.57877002814039807622634211878, 6.59851644814309182136457535503, 6.90539005406248720782903913573, 7.53237436723783915127170820761, 8.790293634045720795055762714128, 9.216262946419771398179803493988