L(s) = 1 | + (−1.48 − 1.48i)3-s + (1.83 − 1.83i)5-s + i·7-s + 1.40i·9-s + (−0.321 + 0.321i)11-s + (4.61 + 4.61i)13-s − 5.45·15-s − 1.84·17-s + (3.88 + 3.88i)19-s + (1.48 − 1.48i)21-s + 5.88i·23-s − 1.74i·25-s + (−2.36 + 2.36i)27-s + (6.14 + 6.14i)29-s − 5.69·31-s + ⋯ |
L(s) = 1 | + (−0.857 − 0.857i)3-s + (0.821 − 0.821i)5-s + 0.377i·7-s + 0.469i·9-s + (−0.0969 + 0.0969i)11-s + (1.28 + 1.28i)13-s − 1.40·15-s − 0.446·17-s + (0.892 + 0.892i)19-s + (0.323 − 0.323i)21-s + 1.22i·23-s − 0.348i·25-s + (−0.454 + 0.454i)27-s + (1.14 + 1.14i)29-s − 1.02·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.421283990\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421283990\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + (1.48 + 1.48i)T + 3iT^{2} \) |
| 5 | \( 1 + (-1.83 + 1.83i)T - 5iT^{2} \) |
| 11 | \( 1 + (0.321 - 0.321i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.61 - 4.61i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 + (-3.88 - 3.88i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.88iT - 23T^{2} \) |
| 29 | \( 1 + (-6.14 - 6.14i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 + (-1.66 + 1.66i)T - 37iT^{2} \) |
| 41 | \( 1 - 10.7iT - 41T^{2} \) |
| 43 | \( 1 + (-0.533 + 0.533i)T - 43iT^{2} \) |
| 47 | \( 1 - 0.465T + 47T^{2} \) |
| 53 | \( 1 + (0.623 - 0.623i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.32 - 7.32i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.57 + 7.57i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.16 - 6.16i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.162iT - 71T^{2} \) |
| 73 | \( 1 + 3.49iT - 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + (-2.51 - 2.51i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.60iT - 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−9.210457619018976884563553518681, −8.670192355943714374142514515548, −7.58385156121086917449043693785, −6.73589344869944652358720273201, −5.98732983441284258754558812632, −5.54973698070507202492466762860, −4.58529370645708524943795960787, −3.34751513261987706092391273529, −1.64144208250377688616988411369, −1.35335802950887183562425368986,
0.64918213909638484583689444616, 2.40009264448939012801636903805, 3.37244372229480359767800418706, 4.40830504253899404138863610723, 5.29365890041315623588962543640, 6.02146612698957774285346225127, 6.56599239747425517291142714127, 7.63309029038478770988278804946, 8.580489130823265173502558040713, 9.521361175047927416149647536749