L(s) = 1 | + 0.488i·2-s + i·3-s + 1.76·4-s + (2.10 − 0.742i)5-s − 0.488·6-s + 5.10i·7-s + 1.83i·8-s − 9-s + (0.362 + 1.02i)10-s + 1.76i·12-s − 1.81i·13-s − 2.49·14-s + (0.742 + 2.10i)15-s + 2.62·16-s + 0.639i·17-s − 0.488i·18-s + ⋯ |
L(s) = 1 | + 0.345i·2-s + 0.577i·3-s + 0.880·4-s + (0.943 − 0.331i)5-s − 0.199·6-s + 1.93i·7-s + 0.649i·8-s − 0.333·9-s + (0.114 + 0.325i)10-s + 0.508i·12-s − 0.504i·13-s − 0.666·14-s + (0.191 + 0.544i)15-s + 0.656·16-s + 0.155i·17-s − 0.115i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.615599612\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.615599612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.10 + 0.742i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.488iT - 2T^{2} \) |
| 7 | \( 1 - 5.10iT - 7T^{2} \) |
| 13 | \( 1 + 1.81iT - 13T^{2} \) |
| 17 | \( 1 - 0.639iT - 17T^{2} \) |
| 19 | \( 1 - 4.39T + 19T^{2} \) |
| 23 | \( 1 - 4.37iT - 23T^{2} \) |
| 29 | \( 1 + 7.56T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 + 5.97iT - 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 3.38iT - 43T^{2} \) |
| 47 | \( 1 - 1.51iT - 47T^{2} \) |
| 53 | \( 1 + 9.17iT - 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 - 7.25iT - 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 - 7.14iT - 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + 0.161iT - 83T^{2} \) |
| 89 | \( 1 - 1.19T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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
−9.495224745661513395716743114922, −8.772504234624591110982241999427, −8.081165899209960245248990501687, −7.03712167952860646046840119470, −5.93439259523699683446413618064, −5.60002315653318193323984834890, −5.11618848570331423661365510777, −3.39815226692194168889993111796, −2.52657393978523246905948140447, −1.75042300944733456025351745115,
0.969952696121216928943480464247, 1.77834108062293654663903959067, 2.89646247068766632034146843986, 3.76008553874661342003219433723, 4.94148399668844487863431557581, 6.16396449775720612297013833012, 6.70864058992259144165799541917, 7.31359313135148059946687565782, 7.897013739587607276668251135872, 9.289266526663455437332106635634