L(s) = 1 | + (2.40 + 1.78i)3-s + 12.2i·7-s + (2.60 + 8.61i)9-s + 8.79i·11-s − 6.20i·13-s − 16.6·17-s − 8.93·19-s + (−21.8 + 29.3i)21-s − 40.1·23-s + (−9.14 + 25.4i)27-s + 4.94i·29-s + 50.6·31-s + (−15.7 + 21.1i)33-s − 41.3i·37-s + (11.0 − 14.9i)39-s + ⋯ |
L(s) = 1 | + (0.802 + 0.596i)3-s + 1.74i·7-s + (0.288 + 0.957i)9-s + 0.799i·11-s − 0.477i·13-s − 0.977·17-s − 0.470·19-s + (−1.03 + 1.39i)21-s − 1.74·23-s + (−0.338 + 0.940i)27-s + 0.170i·29-s + 1.63·31-s + (−0.476 + 0.641i)33-s − 1.11i·37-s + (0.284 − 0.382i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.714543162\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714543162\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.40 - 1.78i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 12.2iT - 49T^{2} \) |
| 11 | \( 1 - 8.79iT - 121T^{2} \) |
| 13 | \( 1 + 6.20iT - 169T^{2} \) |
| 17 | \( 1 + 16.6T + 289T^{2} \) |
| 19 | \( 1 + 8.93T + 361T^{2} \) |
| 23 | \( 1 + 40.1T + 529T^{2} \) |
| 29 | \( 1 - 4.94iT - 841T^{2} \) |
| 31 | \( 1 - 50.6T + 961T^{2} \) |
| 37 | \( 1 + 41.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 10.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 39.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 69.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 96.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 34.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 70.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 31.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 71.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 108. iT - 9.40e3T^{2} \) |
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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
−9.786481989472581946289651850263, −9.090202538525478573993817868233, −8.443345100469032526613751069907, −7.80660314608511620212272740735, −6.54910109186868074487338922878, −5.63183915769795693360899984686, −4.75538227228500729897952414238, −3.84473241115161232716107330052, −2.49522732202487604926187570209, −2.13435371618415339177948181483,
0.42677612076929038535500418713, 1.56431607400577430531435444648, 2.82239992628823314944857826065, 3.95747993252152046100813282963, 4.43730580553731102216237749473, 6.25638616925123286514191022588, 6.64844270908186389408805562863, 7.69885127964458238455160166147, 8.155079628722158746240892166921, 9.077815533259898022767573768765