L(s) = 1 | + (1.14 + 0.832i)2-s + i·3-s + (0.612 + 1.90i)4-s + 1.03i·5-s + (−0.832 + 1.14i)6-s − 3.71·7-s + (−0.885 + 2.68i)8-s − 9-s + (−0.862 + 1.18i)10-s − 1.98i·11-s + (−1.90 + 0.612i)12-s + 4.05i·13-s + (−4.24 − 3.09i)14-s − 1.03·15-s + (−3.24 + 2.33i)16-s + 0.0352·17-s + ⋯ |
L(s) = 1 | + (0.808 + 0.588i)2-s + 0.577i·3-s + (0.306 + 0.951i)4-s + 0.463i·5-s + (−0.340 + 0.466i)6-s − 1.40·7-s + (−0.313 + 0.949i)8-s − 0.333·9-s + (−0.272 + 0.374i)10-s − 0.597i·11-s + (−0.549 + 0.176i)12-s + 1.12i·13-s + (−1.13 − 0.826i)14-s − 0.267·15-s + (−0.812 + 0.583i)16-s + 0.00855·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.267657 + 1.66623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.267657 + 1.66623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 - 0.832i)T \) |
| 3 | \( 1 - iT \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 11 | \( 1 + 1.98iT - 11T^{2} \) |
| 13 | \( 1 - 4.05iT - 13T^{2} \) |
| 17 | \( 1 - 0.0352T + 17T^{2} \) |
| 19 | \( 1 + 0.839iT - 19T^{2} \) |
| 29 | \( 1 - 8.98iT - 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 1.63iT - 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 2.17iT - 43T^{2} \) |
| 47 | \( 1 - 9.26T + 47T^{2} \) |
| 53 | \( 1 - 9.53iT - 53T^{2} \) |
| 59 | \( 1 + 0.521iT - 59T^{2} \) |
| 61 | \( 1 + 1.08iT - 61T^{2} \) |
| 67 | \( 1 + 3.82iT - 67T^{2} \) |
| 71 | \( 1 + 0.0570T + 71T^{2} \) |
| 73 | \( 1 - 2.11T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 8.32T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−11.20074451048449270156174625796, −10.41599831917366606228624321868, −9.248683724208484336908280280787, −8.653240633191354030400753569637, −7.17060061414124652651411631057, −6.59719803820218203716526685954, −5.74563914493580360834131596374, −4.55905444467285048497367368163, −3.51121997330499279226171360891, −2.77936121136013560908421561812,
0.73671851182883926614358081645, 2.45696218973465971346374066656, 3.40126898362482850979704866173, 4.65498320773389528787313153788, 5.78822684747019396045101453813, 6.48379602248750331299020227678, 7.48733914607884785205393412049, 8.754756677459551814100582685516, 9.862390111474163927776772433127, 10.27899957116710779940044934660