L(s) = 1 | + (0.545 + 1.30i)2-s + (−1.40 + 1.42i)4-s − 0.698i·5-s + (−2.62 − 1.05i)8-s + (0.911 − 0.380i)10-s + 2.55·11-s − 1.88·13-s + (−0.0532 − 3.99i)16-s − 3.97i·17-s − 7.05i·19-s + (0.994 + 0.980i)20-s + (1.39 + 3.32i)22-s − 4.02·23-s + 4.51·25-s + (−1.02 − 2.45i)26-s + ⋯ |
L(s) = 1 | + (0.385 + 0.922i)2-s + (−0.702 + 0.711i)4-s − 0.312i·5-s + (−0.927 − 0.373i)8-s + (0.288 − 0.120i)10-s + 0.769·11-s − 0.521·13-s + (−0.0133 − 0.999i)16-s − 0.963i·17-s − 1.61i·19-s + (0.222 + 0.219i)20-s + (0.296 + 0.709i)22-s − 0.839·23-s + 0.902·25-s + (−0.201 − 0.481i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.552795626\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.552795626\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.545 - 1.30i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.698iT - 5T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 + 1.88T + 13T^{2} \) |
| 17 | \( 1 + 3.97iT - 17T^{2} \) |
| 19 | \( 1 + 7.05iT - 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 + 1.86iT - 29T^{2} \) |
| 31 | \( 1 - 0.941iT - 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 3.97iT - 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 - 0.529iT - 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 2.72iT - 67T^{2} \) |
| 71 | \( 1 - 3.51T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 11.8iT - 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.020285461360826196950294630734, −8.594659310103544118847488217593, −7.36449006619183708026870543738, −7.03367861989983422488173306047, −6.10034544024647216887609021841, −5.12401245211148063707715298471, −4.58045956699554498205347436418, −3.57071786012515292207364137591, −2.45442206086270944200145840891, −0.54799028107928627436316676702,
1.31734600330951595961819512338, 2.28260199480671556929414364391, 3.50707732474794604233987938825, 4.03939988362878674181142103788, 5.12986805304008628204968530752, 6.00326924682747856117958821155, 6.70932608539108138098702153882, 7.926042224574181275948476637616, 8.670227786975697402387146882923, 9.537709925796963240048372539407