| L(s) = 1 | + 2.44·5-s + 3.46i·7-s − 2.82i·11-s + 3.46i·13-s − 1.41i·17-s + 4·19-s + 4.89·23-s + 0.999·25-s − 2.44·29-s − 3.46i·31-s + 8.48i·35-s − 1.41i·41-s − 8·43-s − 4.89·47-s − 4.99·49-s + ⋯ |
| L(s) = 1 | + 1.09·5-s + 1.30i·7-s − 0.852i·11-s + 0.960i·13-s − 0.342i·17-s + 0.917·19-s + 1.02·23-s + 0.199·25-s − 0.454·29-s − 0.622i·31-s + 1.43i·35-s − 0.220i·41-s − 1.21·43-s − 0.714·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47954 + 0.324428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47954 + 0.324428i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 7.34T + 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 14.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
| 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
−11.76051063788448163998405914676, −11.11954924188916598969973820676, −9.643825057272424780572489563630, −9.262121100648927874688149482586, −8.243470306468165101096091355078, −6.74458829304317199821371905847, −5.82690038813548006763938626442, −5.04799983282281827033846767857, −3.14539481781211993481595235758, −1.88993731868478107663922835759,
1.42902529885046489985179249064, 3.16592346264426501191292936805, 4.63907974551564495816861100585, 5.70645063116823257688276272452, 6.93124163591534699241382558603, 7.69416119758337514747717904456, 9.081700497899091324681142273693, 10.13575428383238558497675453990, 10.42182900235611355420485417946, 11.70715420212169483887463540308
