L(s) = 1 | − 0.269i·2-s − 3-s + 1.92·4-s − 3.43i·5-s + 0.269i·6-s − 1.97i·7-s − 1.06i·8-s + 9-s − 0.927·10-s − 5.22·11-s − 1.92·12-s − 0.533·13-s − 0.533·14-s + 3.43i·15-s + 3.56·16-s − 2.09·17-s + ⋯ |
L(s) = 1 | − 0.190i·2-s − 0.577·3-s + 0.963·4-s − 1.53i·5-s + 0.110i·6-s − 0.746i·7-s − 0.374i·8-s + 0.333·9-s − 0.293·10-s − 1.57·11-s − 0.556·12-s − 0.147·13-s − 0.142·14-s + 0.886i·15-s + 0.891·16-s − 0.508·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.566176 - 1.04614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.566176 - 1.04614i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 157 | \( 1 + (-6.85 + 10.4i)T \) |
good | 2 | \( 1 + 0.269iT - 2T^{2} \) |
| 5 | \( 1 + 3.43iT - 5T^{2} \) |
| 7 | \( 1 + 1.97iT - 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 0.533T + 13T^{2} \) |
| 17 | \( 1 + 2.09T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 - 6.53iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 9.35T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 8.72iT - 41T^{2} \) |
| 43 | \( 1 + 9.75iT - 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 11.9iT - 53T^{2} \) |
| 59 | \( 1 - 8.86iT - 59T^{2} \) |
| 61 | \( 1 - 15.3iT - 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 - 5.06T + 71T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + 1.66iT - 79T^{2} \) |
| 83 | \( 1 - 1.76iT - 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 1.25iT - 97T^{2} \) |
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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
−10.54933001597025768234389914744, −10.23590950963810721392200495216, −8.912410467392071961798798203822, −7.84199662184792545855650416375, −7.15808028723649705926827447816, −5.81015843545247684325267907430, −5.08993420243733088765008309121, −3.96000278989574486680390147895, −2.22201254562933137128204849208, −0.74054186899590103517039732995,
2.40209850609199749176398085518, 2.92660717743251473084291199349, 4.87073048251130179155930990649, 6.04715254848646559836181071138, 6.56955776453147892902511154179, 7.47251015129704860618325192689, 8.340368913083996937702232064177, 9.966071733453935603943101161153, 10.72564767003850070397673363370, 11.01738746412959380979227543918