L(s) = 1 | + 2.42i·2-s + 1.05i·3-s − 3.89·4-s + (0.123 + 2.23i)5-s − 2.56·6-s − 4.52i·7-s − 4.59i·8-s + 1.88·9-s + (−5.41 + 0.299i)10-s + 0.194·11-s − 4.11i·12-s − 3.48i·13-s + 10.9·14-s + (−2.35 + 0.130i)15-s + 3.36·16-s − 6.60i·17-s + ⋯ |
L(s) = 1 | + 1.71i·2-s + 0.610i·3-s − 1.94·4-s + (0.0551 + 0.998i)5-s − 1.04·6-s − 1.71i·7-s − 1.62i·8-s + 0.627·9-s + (−1.71 + 0.0946i)10-s + 0.0586·11-s − 1.18i·12-s − 0.967i·13-s + 2.93·14-s + (−0.609 + 0.0336i)15-s + 0.840·16-s − 1.60i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0551 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.499759074\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.499759074\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.123 - 2.23i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 2.42iT - 2T^{2} \) |
| 3 | \( 1 - 1.05iT - 3T^{2} \) |
| 7 | \( 1 + 4.52iT - 7T^{2} \) |
| 11 | \( 1 - 0.194T + 11T^{2} \) |
| 13 | \( 1 + 3.48iT - 13T^{2} \) |
| 17 | \( 1 + 6.60iT - 17T^{2} \) |
| 23 | \( 1 + 1.18iT - 23T^{2} \) |
| 29 | \( 1 - 4.10T + 29T^{2} \) |
| 31 | \( 1 - 5.06T + 31T^{2} \) |
| 37 | \( 1 - 4.20iT - 37T^{2} \) |
| 41 | \( 1 - 9.20T + 41T^{2} \) |
| 43 | \( 1 + 2.26iT - 43T^{2} \) |
| 47 | \( 1 + 5.95iT - 47T^{2} \) |
| 53 | \( 1 + 6.94iT - 53T^{2} \) |
| 59 | \( 1 + 2.39T + 59T^{2} \) |
| 61 | \( 1 - 4.79T + 61T^{2} \) |
| 67 | \( 1 - 5.22iT - 67T^{2} \) |
| 71 | \( 1 - 5.85T + 71T^{2} \) |
| 73 | \( 1 + 6.54iT - 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 - 9.54iT - 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.41iT - 97T^{2} \) |
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
−9.610029164952597883888131541404, −8.319055789407855587010414993292, −7.61876280612403175309136645956, −6.99635073032953733485595521925, −6.68616575615054839083480154370, −5.51332135402376472772524117915, −4.64515072586520678669754167444, −4.02988594957135666991166392197, −2.99169901804282538851474466468, −0.65522169545966561239039336926,
1.19966783384489086758989262305, 1.89601677706647067926606343811, 2.63967160706497215656143714603, 4.02720467621738823598795417949, 4.60004312824008711520241561451, 5.69684637655808469915745739637, 6.45501510351986445909281133648, 7.952146988855053206560631639855, 8.597342302102933602247147305617, 9.245934082178104279417098616209