| L(s) = 1 | + (−1.40 + 0.178i)2-s + (1.93 − 0.5i)4-s − 0.356i·5-s − 7-s + (−2.62 + 1.04i)8-s + (0.0635 + 0.5i)10-s + 5.96i·11-s − 2.87i·13-s + (1.40 − 0.178i)14-s + (3.50 − 1.93i)16-s + 3.16·17-s − 0.127i·19-s + (−0.178 − 0.690i)20-s + (−1.06 − 8.37i)22-s − 2.80·23-s + ⋯ |
| L(s) = 1 | + (−0.992 + 0.126i)2-s + (0.968 − 0.250i)4-s − 0.159i·5-s − 0.377·7-s + (−0.929 + 0.370i)8-s + (0.0200 + 0.158i)10-s + 1.79i·11-s − 0.796i·13-s + (0.374 − 0.0476i)14-s + (0.875 − 0.484i)16-s + 0.766·17-s − 0.0291i·19-s + (−0.0398 − 0.154i)20-s + (−0.226 − 1.78i)22-s − 0.585·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6543078043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6543078043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.178i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 0.356iT - 5T^{2} \) |
| 11 | \( 1 - 5.96iT - 11T^{2} \) |
| 13 | \( 1 + 2.87iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 0.127iT - 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 - 2.44iT - 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 + 1.12iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6.63iT - 59T^{2} \) |
| 61 | \( 1 - 13.7iT - 61T^{2} \) |
| 67 | \( 1 - 8.61iT - 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 - 0.872T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 6.63iT - 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 + 9.74T + 97T^{2} \) |
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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
−9.846252765951989900833548914111, −8.967070069030996890688796376025, −8.148871872340727985281044835524, −7.31562536161445384232267141711, −6.84224174662846694616645303917, −5.72266725076918277199354173664, −4.91389155694979827922162425628, −3.54124683492880151760624993694, −2.43218289155926521795765659996, −1.31331939543842238212354345425,
0.37185068443447219278213697201, 1.75881668261124496464759877801, 3.08573186008268237773925384356, 3.67337520973268467818594404088, 5.32303031970286095931383269046, 6.24065832289620808877013920626, 6.79089774458659360370138931969, 7.87962418349666567116217899830, 8.446259967772142060290573441636, 9.256868460622593172946743797456
