L(s) = 1 | − i·3-s − 5-s + 4.47·7-s + 2·9-s − 3i·11-s + 13-s + i·15-s − 4.47i·17-s + 4i·19-s − 4.47i·21-s + 4.47·23-s − 4·25-s − 5i·27-s + (3 + 4.47i)29-s + 5i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447·5-s + 1.69·7-s + 0.666·9-s − 0.904i·11-s + 0.277·13-s + 0.258i·15-s − 1.08i·17-s + 0.917i·19-s − 0.975i·21-s + 0.932·23-s − 0.800·25-s − 0.962i·27-s + (0.557 + 0.830i)29-s + 0.898i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.156720112\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.156720112\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (-3 - 4.47i)T \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 8.94iT - 73T^{2} \) |
| 79 | \( 1 - 15iT - 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 13.4iT - 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−8.778037233739535162727708615895, −8.310243529383239306640141222226, −7.51858842580402130956155962917, −7.03443261374733835515707021697, −5.85934883382914454728278371033, −5.01548351931013075866145275791, −4.25410893852904764719896235974, −3.15058533608362532325851652377, −1.80645496015045977964939515932, −0.971158506903097112317827998623,
1.28999199747980159103584081060, 2.29563519122987647899978825162, 3.83964071288135945493792110630, 4.48599381485647143423684282332, 4.96119961592316698842263259113, 6.10442342874628735167884510683, 7.29413593535235948786089244216, 7.72533688668088961363755364575, 8.588579718055368047586563830403, 9.298048171558416843681043185997