L(s) = 1 | + 2.31i·2-s + 2.90i·3-s − 3.35·4-s + (1.84 − 1.26i)5-s − 6.72·6-s − 2.34i·7-s − 3.14i·8-s − 5.43·9-s + (2.92 + 4.26i)10-s − 3.45·11-s − 9.74i·12-s − 3.29i·13-s + 5.43·14-s + (3.67 + 5.35i)15-s + 0.554·16-s − 1.38i·17-s + ⋯ |
L(s) = 1 | + 1.63i·2-s + 1.67i·3-s − 1.67·4-s + (0.824 − 0.565i)5-s − 2.74·6-s − 0.886i·7-s − 1.11i·8-s − 1.81·9-s + (0.926 + 1.34i)10-s − 1.04·11-s − 2.81i·12-s − 0.912i·13-s + 1.45·14-s + (0.948 + 1.38i)15-s + 0.138·16-s − 0.335i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6894143023\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6894143023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.84 + 1.26i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.31iT - 2T^{2} \) |
| 3 | \( 1 - 2.90iT - 3T^{2} \) |
| 7 | \( 1 + 2.34iT - 7T^{2} \) |
| 11 | \( 1 + 3.45T + 11T^{2} \) |
| 13 | \( 1 + 3.29iT - 13T^{2} \) |
| 17 | \( 1 + 1.38iT - 17T^{2} \) |
| 23 | \( 1 + 8.90iT - 23T^{2} \) |
| 29 | \( 1 + 6.28T + 29T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 - 0.650iT - 37T^{2} \) |
| 41 | \( 1 - 2.01T + 41T^{2} \) |
| 43 | \( 1 + 1.72iT - 43T^{2} \) |
| 47 | \( 1 - 6.32iT - 47T^{2} \) |
| 53 | \( 1 - 2.12iT - 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 + 2.46iT - 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 6.88iT - 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 1.84iT - 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 + 7.61iT - 97T^{2} \) |
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
−9.193624321263350486978877220808, −8.580299254901305534144602040009, −7.83979308720155290421652842189, −6.95238698690841278356003312287, −5.76650058829523922105554630377, −5.43817832104994140763245183398, −4.67090305905614074448002690306, −4.04826157389403517254781114657, −2.70397196583524763299025160829, −0.24019316393194579885862825516,
1.54300290347935757630190411638, 2.02680421317413311491091149655, 2.68600582600942100953694334951, 3.65493224761963499935815298736, 5.35613227233981643088585931540, 5.81536636175508035661149122761, 6.94130319180305493677597946789, 7.56102361834047121081196218070, 8.679352121527169787045792165558, 9.298037334642535395057334274865