L(s) = 1 | + 2i·3-s − 9-s − 11-s + 2i·13-s + 4i·17-s − 5i·23-s + 4i·27-s + 3·29-s − 10·31-s − 2i·33-s + 5i·37-s − 4·39-s − 10·41-s + 5i·43-s − 4i·47-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 0.333·9-s − 0.301·11-s + 0.554i·13-s + 0.970i·17-s − 1.04i·23-s + 0.769i·27-s + 0.557·29-s − 1.79·31-s − 0.348i·33-s + 0.821i·37-s − 0.640·39-s − 1.56·41-s + 0.762i·43-s − 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7875716288\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7875716288\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 5iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 5iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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
−8.479780481918265994114772981061, −8.396919150111542506385287243097, −7.06341236442745217485343518751, −6.60687228164933156603825049216, −5.50132617461632731064280111727, −5.00850408712891194826939959623, −4.08837257644761070241174294629, −3.68359321194566141461283702519, −2.59047335427488914792070474290, −1.51409863939534245663434146025,
0.20949553402196277461448617906, 1.32395086792341645603887287922, 2.19955543502290383046998659232, 3.08604894330988074959902592730, 4.00657304398252802592430232357, 5.16947121174559560350480550373, 5.63928315415110487195383417100, 6.58761326160545569996550848008, 7.30070816748168728670874731436, 7.57315257699442388065483324722