L(s) = 1 | + (−0.707 + 0.707i)5-s + 2.87·7-s + (2.03 + 2.03i)11-s + (3.87 − 3.87i)13-s + 1.41i·17-s + (−5.87 − 5.87i)19-s − 5.47i·23-s − 1.00i·25-s + (2.82 + 2.82i)29-s − 4i·31-s + (−2.03 + 2.03i)35-s + (−1 − i)37-s + 11.1·41-s + (6 − 6i)43-s + 1.41·47-s + ⋯ |
L(s) = 1 | + (−0.316 + 0.316i)5-s + 1.08·7-s + (0.612 + 0.612i)11-s + (1.07 − 1.07i)13-s + 0.342i·17-s + (−1.34 − 1.34i)19-s − 1.14i·23-s − 0.200i·25-s + (0.525 + 0.525i)29-s − 0.718i·31-s + (−0.343 + 0.343i)35-s + (−0.164 − 0.164i)37-s + 1.73·41-s + (0.914 − 0.914i)43-s + 0.206·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.059875883\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.059875883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - 2.87T + 7T^{2} \) |
| 11 | \( 1 + (-2.03 - 2.03i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.87 + 3.87i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 + (5.87 + 5.87i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.47iT - 23T^{2} \) |
| 29 | \( 1 + (-2.82 - 2.82i)T + 29iT^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.10 + 9.10i)T + 59iT^{2} \) |
| 61 | \( 1 + (10.7 - 10.7i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.12 + 3.12i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.71iT - 71T^{2} \) |
| 73 | \( 1 - 5.74iT - 73T^{2} \) |
| 79 | \( 1 - 3.74iT - 79T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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
−8.619244982491552463316452287087, −8.033964117915184733909423472127, −7.24616626398931223128309487835, −6.43540900402338420092317700714, −5.71573955218183031088489764156, −4.52429493867201459436004574099, −4.22397659285220619895880960779, −2.95633854441634511594720804089, −2.00311343388849263138044509728, −0.76176331537755413747814468749,
1.18627094243573238726904594675, 1.89667546280442125936836157188, 3.39407296905784386325376460019, 4.19101976952161684567937483444, 4.74958434788598620229154740004, 6.00820646797897298290244756034, 6.30379292354991180911637236363, 7.61543031654954062986832122221, 8.025306579228804452083220230079, 8.930910978658644563941681688263