L(s) = 1 | + (0.629 + 1.26i)2-s + (−1.20 + 1.59i)4-s + 2.32i·5-s + (−2.77 − 0.524i)8-s + (−2.94 + 1.46i)10-s + 3.58i·11-s + 2.93i·13-s + (−1.08 − 3.84i)16-s + 2.32i·17-s + 8.33·19-s + (−3.71 − 2.80i)20-s + (−4.53 + 2.25i)22-s + 1.48i·23-s − 0.414·25-s + (−3.71 + 1.84i)26-s + ⋯ |
L(s) = 1 | + (0.445 + 0.895i)2-s + (−0.603 + 0.797i)4-s + 1.04i·5-s + (−0.982 − 0.185i)8-s + (−0.931 + 0.463i)10-s + 1.07i·11-s + 0.812i·13-s + (−0.271 − 0.962i)16-s + 0.564i·17-s + 1.91·19-s + (−0.829 − 0.628i)20-s + (−0.966 + 0.480i)22-s + 0.309i·23-s − 0.0828·25-s + (−0.727 + 0.361i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.574427471\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.574427471\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.629 - 1.26i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2.32iT - 5T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 2.32iT - 17T^{2} \) |
| 19 | \( 1 - 8.33T + 19T^{2} \) |
| 23 | \( 1 - 1.48iT - 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 2.89iT - 41T^{2} \) |
| 43 | \( 1 + 6.37iT - 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.93iT - 61T^{2} \) |
| 67 | \( 1 + 9.02iT - 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 15.3iT - 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 13.5iT - 89T^{2} \) |
| 97 | \( 1 - 5.35iT - 97T^{2} \) |
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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
−9.487276197472208843178708098662, −9.053126642953072670189908623218, −7.74512884500019815057825524686, −7.26006490427896295169244700706, −6.76691891339819315234837276462, −5.75044200631289104017080828867, −5.02346166929599012848377685350, −3.91397213395005154718189124130, −3.25205199122049381644273551790, −1.95948837231462828078858027507,
0.52535952304816477550715255870, 1.45729274791901545824670615685, 2.95282673667154979803073388281, 3.57446334539842956756018566850, 4.78376604296081880142229882179, 5.38567346992501183379735471215, 5.97230173701373003112002441326, 7.39193772846249967901715334808, 8.255430348505038493432912664416, 9.168054185986323628085785300663