L(s) = 1 | + i·3-s − 1.52i·7-s − 9-s − 3.59·11-s + 5.59i·13-s − 4.07i·17-s + 1.59·19-s + 1.52·21-s + i·23-s − i·27-s + 4.07·29-s + 2.47·31-s − 3.59i·33-s − 9.66i·37-s − 5.59·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.576i·7-s − 0.333·9-s − 1.08·11-s + 1.55i·13-s − 0.987i·17-s + 0.366·19-s + 0.333·21-s + 0.208i·23-s − 0.192i·27-s + 0.756·29-s + 0.444·31-s − 0.626i·33-s − 1.58i·37-s − 0.896·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.205521536\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.205521536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + 1.52iT - 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.59iT - 13T^{2} \) |
| 17 | \( 1 + 4.07iT - 17T^{2} \) |
| 19 | \( 1 - 1.59T + 19T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 9.66iT - 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 7.05iT - 43T^{2} \) |
| 47 | \( 1 - 4.54iT - 47T^{2} \) |
| 53 | \( 1 - 5.01iT - 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + 2.47iT - 67T^{2} \) |
| 71 | \( 1 + 5.01T + 71T^{2} \) |
| 73 | \( 1 + 8.79iT - 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 - 9.12iT - 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 13.0iT - 97T^{2} \) |
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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
−8.105982591166015804382736796060, −7.48021543833518073933753514348, −6.88805791575629520669677090682, −6.03497269451251939310928572123, −5.23465999186054543805183626634, −4.53106480327927152635633775959, −4.03439670222630577315164458944, −2.97620399042159755398151797308, −2.30160904689312080271943974727, −0.989565487891282123730398187473,
0.33635697090319372187821031634, 1.46312776436610417168769319279, 2.63108951382426878518205432549, 2.96445436767172907508465024152, 4.08839807211199519017279939242, 5.21905450101991120092066136182, 5.53214473893523250768828296274, 6.29154662065660759765049929273, 7.07802191378314307797549108144, 7.904550227898408496517521352353