L(s) = 1 | + 3-s + (−2.11 − 0.726i)5-s + 4.05i·7-s + 9-s + 0.985i·11-s + 4.94·13-s + (−2.11 − 0.726i)15-s + 4.52i·17-s + 2.60i·19-s + 4.05i·21-s + 3.53i·23-s + (3.94 + 3.07i)25-s + 27-s − 7.59i·29-s + 3.28·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.945 − 0.324i)5-s + 1.53i·7-s + 0.333·9-s + 0.297i·11-s + 1.37·13-s + (−0.546 − 0.187i)15-s + 1.09i·17-s + 0.597i·19-s + 0.885i·21-s + 0.737i·23-s + (0.789 + 0.614i)25-s + 0.192·27-s − 1.41i·29-s + 0.589·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27706 + 0.706889i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27706 + 0.706889i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (2.11 + 0.726i)T \) |
good | 7 | \( 1 - 4.05iT - 7T^{2} \) |
| 11 | \( 1 - 0.985iT - 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 4.52iT - 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 - 3.53iT - 23T^{2} \) |
| 29 | \( 1 + 7.59iT - 29T^{2} \) |
| 31 | \( 1 - 3.28T + 31T^{2} \) |
| 37 | \( 1 + 0.945T + 37T^{2} \) |
| 41 | \( 1 - 0.568T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 2.60iT - 47T^{2} \) |
| 53 | \( 1 - 0.229T + 53T^{2} \) |
| 59 | \( 1 + 9.10iT - 59T^{2} \) |
| 61 | \( 1 - 11.0iT - 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 1.43T + 71T^{2} \) |
| 73 | \( 1 + 11.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.28T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 3.23iT - 97T^{2} \) |
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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
−11.37023398805462400895477677125, −10.15469986166632064698444129873, −9.079440800244020319266845179690, −8.397914573364896415064845485984, −7.911490082913086618203997188597, −6.44630198777069939182736022673, −5.52535646806503124228940188114, −4.18008633837503964623073510671, −3.25702569882354700098196248490, −1.78018408387523595314253450485,
0.919617039958997677208070014175, 3.06800677854131571684795347737, 3.84981345762096773235078596995, 4.79601001560153968163757467580, 6.61955037002223859956312775688, 7.18484783892715893220936665610, 8.132474107587737710862591966870, 8.861017395279788473925729750022, 10.11789302140291410865896650915, 10.89872432101918738483299629957