| L(s) = 1 | + (1.46 + 0.916i)3-s − 1.83·5-s − i·7-s + (1.31 + 2.69i)9-s + 5.57i·13-s + (−2.69 − 1.68i)15-s + 6.08i·17-s + 6.93·19-s + (0.916 − 1.46i)21-s − 0.697·23-s − 1.63·25-s + (−0.530 + 5.16i)27-s + 7.11·29-s + 1.83i·35-s + 1.87i·37-s + ⋯ |
| L(s) = 1 | + (0.848 + 0.529i)3-s − 0.819·5-s − 0.377i·7-s + (0.439 + 0.898i)9-s + 1.54i·13-s + (−0.695 − 0.433i)15-s + 1.47i·17-s + 1.59·19-s + (0.200 − 0.320i)21-s − 0.145·23-s − 0.327·25-s + (−0.102 + 0.994i)27-s + 1.32·29-s + 0.309i·35-s + 0.308i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32713 + 1.03255i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32713 + 1.03255i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.46 - 0.916i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.83T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.57iT - 13T^{2} \) |
| 17 | \( 1 - 6.08iT - 17T^{2} \) |
| 19 | \( 1 - 6.93T + 19T^{2} \) |
| 23 | \( 1 + 0.697T + 23T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 + 4.69iT - 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 4.30iT - 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 + 0.438iT - 83T^{2} \) |
| 89 | \( 1 + 2.64iT - 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.58249383040139515946087512320, −9.759636986532578285983250304625, −8.951509898954055660444534391197, −8.090123795820785490283900203283, −7.43496544707878537275151412407, −6.38017055963432306673091409112, −4.85732857997018873144706272696, −4.04674965616463584013441953947, −3.29884625288680330731072295097, −1.74235581112556368978813290084,
0.875898040025905573793033105663, 2.75562286562231778517233645325, 3.34121715143151803048464949006, 4.74113048824370531416053304531, 5.85157137256237005584082640631, 7.16123461275504788654591494015, 7.72541792240700070343965472271, 8.399291639477324709908574378153, 9.398753442956138463430400684169, 10.11602857842560233889012390313
