L(s) = 1 | + i·3-s + i·5-s − 4.68·7-s − 9-s + 3.83·11-s − 2.63·13-s − 15-s − 5.43i·17-s + 4.36·19-s − 4.68i·21-s + (−4.47 − 1.71i)23-s − 25-s − i·27-s + 6.33·29-s + 4.22i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 1.76·7-s − 0.333·9-s + 1.15·11-s − 0.731·13-s − 0.258·15-s − 1.31i·17-s + 1.00·19-s − 1.02i·21-s + (−0.933 − 0.357i)23-s − 0.200·25-s − 0.192i·27-s + 1.17·29-s + 0.758i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174185281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174185281\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 23 | \( 1 + (4.47 + 1.71i)T \) |
good | 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 + 5.43iT - 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 - 4.22iT - 31T^{2} \) |
| 37 | \( 1 + 5.00iT - 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 5.14iT - 47T^{2} \) |
| 53 | \( 1 - 2.61iT - 53T^{2} \) |
| 59 | \( 1 - 1.55iT - 59T^{2} \) |
| 61 | \( 1 - 11.4iT - 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 12.0iT - 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 3.50T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−8.570966339339210307180243644093, −7.25243867728329851067979116945, −7.06098109919440875464189401992, −6.18710074313132651937199391299, −5.62193669537407312384524945436, −4.55271658276631764501722341362, −3.84503860413908393223579716295, −3.03325218062673289400060852466, −2.56646965201117672858558356724, −0.829124891620464699039810687028,
0.41224490574950001906979117831, 1.50500518523833353849120067361, 2.57027395182130708757847549299, 3.51506153880787033688962620008, 4.03976767764865179536802497945, 5.18665010815259070975271503072, 6.14469789304046635756266322580, 6.38359229846770043129904538468, 7.14655708614863914782249166584, 7.930602738812864076093703265808