L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.80 − 1.32i)5-s + (0.949 + 0.949i)7-s + (−0.707 − 0.707i)8-s + (−2.21 + 0.339i)10-s − 2.64i·11-s + (2 − 2i)13-s + 1.34·14-s − 1.00·16-s + (4.94 − 4.94i)17-s + 2.42i·19-s + (−1.32 + 1.80i)20-s + (−1.87 − 1.87i)22-s + (0.707 + 0.707i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.806 − 0.591i)5-s + (0.358 + 0.358i)7-s + (−0.250 − 0.250i)8-s + (−0.698 + 0.107i)10-s − 0.797i·11-s + (0.554 − 0.554i)13-s + 0.358·14-s − 0.250·16-s + (1.20 − 1.20i)17-s + 0.555i·19-s + (−0.295 + 0.403i)20-s + (−0.398 − 0.398i)22-s + (0.147 + 0.147i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.689760260\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.689760260\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.80 + 1.32i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (-0.949 - 0.949i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 + (-2 + 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.94 + 4.94i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.42iT - 19T^{2} \) |
| 29 | \( 1 + 0.383T + 29T^{2} \) |
| 31 | \( 1 + 6.99T + 31T^{2} \) |
| 37 | \( 1 + (0.820 + 0.820i)T + 37iT^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 + (2.82 - 2.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.19 - 2.19i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.11 + 7.11i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 + 8.31T + 61T^{2} \) |
| 67 | \( 1 + (2.59 + 2.59i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.76iT - 71T^{2} \) |
| 73 | \( 1 + (-8.26 + 8.26i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 + (5.56 + 5.56i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 + (-4.46 - 4.46i)T + 97iT^{2} \) |
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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
−8.856631001530432120744939839119, −8.000472032366279386057724403741, −7.45093510705469486940092393049, −6.16272830259203575901203530322, −5.36320303604683861071843952593, −4.84711625820911434003448316500, −3.56850451985635202772589056363, −3.22883010522982235134371886660, −1.67694922181872693157774510413, −0.52463770813959139548059000000,
1.55683469688318817998700771690, 2.97568577547585755097217776104, 3.88140786311959763186967310085, 4.43751093834737411114673843304, 5.47024846047285788780003774274, 6.43124054216116194575096464026, 7.09405800934057540341288255905, 7.77081624764695799696740900596, 8.356497429572940653271046972303, 9.343974938530500169573216905717