L(s) = 1 | − i·7-s − 2·11-s − 4i·13-s − 6i·17-s − 6·19-s − 8i·23-s − 2·29-s + 10·31-s + 2i·37-s − 10·41-s + 4i·43-s + 8i·47-s − 49-s + 4i·53-s − 8·59-s + ⋯ |
L(s) = 1 | − 0.377i·7-s − 0.603·11-s − 1.10i·13-s − 1.45i·17-s − 1.37·19-s − 1.66i·23-s − 0.371·29-s + 1.79·31-s + 0.328i·37-s − 1.56·41-s + 0.609i·43-s + 1.16i·47-s − 0.142·49-s + 0.549i·53-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2753284953\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2753284953\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−7.82055382401402016956848164593, −6.78859504872157359077603761929, −6.41033906921793711440143414599, −5.37674179666073660698854757516, −4.76098828910067998060959775398, −4.09519789922521185435075285690, −2.84960237846029342875821363530, −2.56760083606041036107231029371, −1.05135736658835141369752897040, −0.07234495905298095023881329306,
1.65426306973481533607872085843, 2.16895641289312599970610353966, 3.36954068535566271756527601226, 4.04315797698719992973885550375, 4.87022076872457111084152336141, 5.63030206803935319966843218590, 6.40352020443916823903318927364, 6.87101946947701337829544411602, 7.939915264148380675405234112440, 8.346828743525269137659479265272