L(s) = 1 | + (1.41 − i)3-s + (−0.414 − 0.414i)5-s + 7-s + (1.00 − 2.82i)9-s + (−3.82 + 3.82i)11-s + (−4.41 − 4.41i)13-s + (−1 − 0.171i)15-s − 1.17i·17-s + (−0.414 + 0.414i)19-s + (1.41 − i)21-s − 7.65i·23-s − 4.65i·25-s + (−1.41 − 5.00i)27-s + (3 − 3i)29-s − 6.48i·31-s + ⋯ |
L(s) = 1 | + (0.816 − 0.577i)3-s + (−0.185 − 0.185i)5-s + 0.377·7-s + (0.333 − 0.942i)9-s + (−1.15 + 1.15i)11-s + (−1.22 − 1.22i)13-s + (−0.258 − 0.0442i)15-s − 0.284i·17-s + (−0.0950 + 0.0950i)19-s + (0.308 − 0.218i)21-s − 1.59i·23-s − 0.931i·25-s + (−0.272 − 0.962i)27-s + (0.557 − 0.557i)29-s − 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.555498831\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.555498831\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 + i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (0.414 + 0.414i)T + 5iT^{2} \) |
| 11 | \( 1 + (3.82 - 3.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.41 + 4.41i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.17iT - 17T^{2} \) |
| 19 | \( 1 + (0.414 - 0.414i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.65iT - 23T^{2} \) |
| 29 | \( 1 + (-3 + 3i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.48iT - 31T^{2} \) |
| 37 | \( 1 + (-1.82 + 1.82i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 + (-3.82 - 3.82i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \) |
| 59 | \( 1 + (10.0 - 10.0i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.41 - 2.41i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7 + 7i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 - 5.17iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + (-8.89 - 8.89i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.349033666311633233360367722118, −8.170817438184613239119299032477, −7.87759615722013045353248580999, −7.19545466412156150810016797859, −6.12154035371422460768268169268, −4.92864334965224478349472467393, −4.28926946209614655461573189420, −2.66962057092234190026227808924, −2.35934118085590493804656620394, −0.54262937597803777668846791187,
1.81789575827153725628722680339, 2.92040348225390933103447949828, 3.69623030310295847715360077361, 4.87470439890109722823175340107, 5.39434541846945957116655409327, 6.84410911417967824106899033991, 7.62642165503271749913619972414, 8.284135527352705264344096541638, 9.102131190354683852639542040322, 9.763513818149264544784476946695