L(s) = 1 | + (0.260 + 2.22i)5-s + (2.63 + 0.260i)7-s + (3.34 − 1.93i)11-s + (4.35 − 4.35i)13-s + (−4.66 − 1.24i)17-s + (1.57 + 0.907i)19-s + (2.15 − 0.576i)23-s + (−4.86 + 1.15i)25-s − 0.529·29-s + (−3.19 − 5.53i)31-s + (0.106 + 5.91i)35-s + (9.69 − 2.59i)37-s + 5.31i·41-s + (−1.82 + 1.82i)43-s + (2.59 + 9.69i)47-s + ⋯ |
L(s) = 1 | + (0.116 + 0.993i)5-s + (0.995 + 0.0985i)7-s + (1.00 − 0.581i)11-s + (1.20 − 1.20i)13-s + (−1.13 − 0.302i)17-s + (0.360 + 0.208i)19-s + (0.448 − 0.120i)23-s + (−0.972 + 0.231i)25-s − 0.0983·29-s + (−0.573 − 0.993i)31-s + (0.0179 + 0.999i)35-s + (1.59 − 0.427i)37-s + 0.830i·41-s + (−0.277 + 0.277i)43-s + (0.378 + 1.41i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.083637796\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.083637796\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.260 - 2.22i)T \) |
| 7 | \( 1 + (-2.63 - 0.260i)T \) |
good | 11 | \( 1 + (-3.34 + 1.93i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.35 + 4.35i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.66 + 1.24i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.57 - 0.907i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.15 + 0.576i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 0.529T + 29T^{2} \) |
| 31 | \( 1 + (3.19 + 5.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.69 + 2.59i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 5.31iT - 41T^{2} \) |
| 43 | \( 1 + (1.82 - 1.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.59 - 9.69i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.0180 - 0.0673i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.07 + 1.86i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.99 + 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.36 - 5.08i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 8.72iT - 71T^{2} \) |
| 73 | \( 1 + (3.55 + 0.953i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.564 + 0.325i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.54 + 4.54i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7.34 + 12.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.639 - 0.639i)T + 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.697399105998765239228306777370, −8.857412523854371594661860864205, −8.063413697450119619191384656617, −7.32771378710476789511034475633, −6.23250253990716181476678500324, −5.79128240670461209264745101869, −4.46713168498247483455632781340, −3.53662911382177809762174171005, −2.52480511406739298657149668647, −1.15641404061544825315011603757,
1.25438614909209560786997850093, 1.97485671574246686879608533596, 3.90629498848381673085254795951, 4.42361065556895259307473109076, 5.31054100099868852853230613961, 6.39945301640320324541278049046, 7.14730133340819751792726262268, 8.268796853196635941515674324267, 8.959706920811723965126636411612, 9.275790280243917012375197792151