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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3566573304\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3566573304\) |
\(L(\frac{3}{2})\) |
|
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
−9.538657094522965517635629161939, −8.885486227475516337499885675572, −8.107936112052473032060885940930, −7.30353771636194456147934893768, −6.49093997613668673954293079583, −5.07131357531840693749087004126, −4.60966510474871628034246564453, −3.21492442352150722835261838039, −2.70953421739657385612441834866, −1.41948091194728482721637279522,
0.15651073945978104141489081203, 1.64913835470040613686119683269, 3.12206896102899205974375061575, 4.40937088656905650355956835598, 5.19159519471389996852011825522, 5.77355009838084710720775723186, 6.90369361433062386516723477222, 7.45621795635053685971047495082, 8.262819620247881066206988476694, 8.927885199428749453928275395267