L(s) = 1 | + (−1.5 + 0.866i)3-s + (1 − 1.73i)4-s + (1.5 − 2.59i)5-s + (2 + 3.46i)7-s + (1.5 − 2.59i)9-s + (−0.5 − 0.866i)11-s + 3.46i·12-s + (−1 + 1.73i)13-s + 5.19i·15-s + (−1.99 − 3.46i)16-s − 6·17-s + 2·19-s + (−3 − 5.19i)20-s + (−6 − 3.46i)21-s + (−1.5 + 2.59i)23-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (0.5 − 0.866i)4-s + (0.670 − 1.16i)5-s + (0.755 + 1.30i)7-s + (0.5 − 0.866i)9-s + (−0.150 − 0.261i)11-s + 0.999i·12-s + (−0.277 + 0.480i)13-s + 1.34i·15-s + (−0.499 − 0.866i)16-s − 1.45·17-s + 0.458·19-s + (−0.670 − 1.16i)20-s + (−1.30 − 0.755i)21-s + (−0.312 + 0.541i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.959632 - 0.169209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.959632 - 0.169209i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9 - 15.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.92475801639020779272545550253, −12.50295702500184052377755686661, −11.67494008311005808829522411419, −10.76853697888891159864304747996, −9.454431631110923038638750997477, −8.799610798073646460970337728645, −6.59449577592498389311843276610, −5.37548419439968430942839636954, −4.98208456973095822275834559813, −1.78227959678888008814466134459,
2.33353216978153844795617437488, 4.38376850482268117782348579761, 6.27507846626751700891588224654, 7.14261781172656268895974449960, 7.901228967415635956236179770929, 10.14676737573860952850416345980, 10.94559451476732872807593857280, 11.59036606723586378882373788205, 13.03249938638593505731519680935, 13.68456216274581448742508932732