L(s) = 1 | + (1.39 − 1.02i)3-s + 1.12i·5-s − 3.21·7-s + (0.886 − 2.86i)9-s − 3.06i·11-s − 0.110i·13-s + (1.15 + 1.56i)15-s − 1.89i·17-s + (2.66 − 3.45i)19-s + (−4.48 + 3.30i)21-s − 7.89i·23-s + 3.73·25-s + (−1.71 − 4.90i)27-s − 1.77·29-s − 5.07i·31-s + ⋯ |
L(s) = 1 | + (0.804 − 0.593i)3-s + 0.502i·5-s − 1.21·7-s + (0.295 − 0.955i)9-s − 0.925i·11-s − 0.0306i·13-s + (0.298 + 0.404i)15-s − 0.460i·17-s + (0.610 − 0.791i)19-s + (−0.978 + 0.721i)21-s − 1.64i·23-s + 0.747·25-s + (−0.329 − 0.944i)27-s − 0.329·29-s − 0.911i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0215 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0215 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13796 - 1.16271i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13796 - 1.16271i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.39 + 1.02i)T \) |
| 19 | \( 1 + (-2.66 + 3.45i)T \) |
good | 5 | \( 1 - 1.12iT - 5T^{2} \) |
| 7 | \( 1 + 3.21T + 7T^{2} \) |
| 11 | \( 1 + 3.06iT - 11T^{2} \) |
| 13 | \( 1 + 0.110iT - 13T^{2} \) |
| 17 | \( 1 + 1.89iT - 17T^{2} \) |
| 23 | \( 1 + 7.89iT - 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 + 5.07iT - 31T^{2} \) |
| 37 | \( 1 - 3.59iT - 37T^{2} \) |
| 41 | \( 1 - 1.01T + 41T^{2} \) |
| 43 | \( 1 - 8.31T + 43T^{2} \) |
| 47 | \( 1 - 5.68iT - 47T^{2} \) |
| 53 | \( 1 - 0.535T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 2.76T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 - 17.5iT - 79T^{2} \) |
| 83 | \( 1 + 8.67iT - 83T^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−9.673987301315312470855562764609, −9.070480906513144426354335507211, −8.241726235142930615235055741880, −7.21206574959027565891066527616, −6.61474488600082381106067311885, −5.84393406455390507556499136682, −4.25741969924202448060628084990, −3.04301228004406467764099310199, −2.68879618116951309500498719687, −0.70582228707070036097261147945,
1.72140823064859246616010088569, 3.11693673828595407411233480764, 3.84308053570042253467160338746, 4.91286959096148731017847587880, 5.87026137197600497917106887473, 7.11772597793490803856290935951, 7.78530305470037332468347363836, 8.913850939043940013566395626881, 9.441884228697117295494103178542, 10.04213143878808192639332091936