| L(s) = 1 | + (−1.04 − 0.947i)2-s + (0.664 − 0.791i)3-s + (0.204 + 1.98i)4-s + (−0.681 − 2.12i)5-s + (−1.44 + 0.201i)6-s + (1.98 + 3.43i)7-s + (1.67 − 2.28i)8-s + (0.335 + 1.90i)9-s + (−1.30 + 2.88i)10-s + (3.95 + 2.28i)11-s + (1.71 + 1.16i)12-s + (1.59 − 1.34i)13-s + (1.17 − 5.48i)14-s + (−2.13 − 0.875i)15-s + (−3.91 + 0.812i)16-s + (3.02 + 0.532i)17-s + ⋯ |
| L(s) = 1 | + (−0.742 − 0.670i)2-s + (0.383 − 0.457i)3-s + (0.102 + 0.994i)4-s + (−0.304 − 0.952i)5-s + (−0.591 + 0.0823i)6-s + (0.749 + 1.29i)7-s + (0.590 − 0.806i)8-s + (0.111 + 0.634i)9-s + (−0.411 + 0.911i)10-s + (1.19 + 0.689i)11-s + (0.493 + 0.334i)12-s + (0.442 − 0.371i)13-s + (0.313 − 1.46i)14-s + (−0.552 − 0.225i)15-s + (−0.979 + 0.203i)16-s + (0.732 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.09188 - 0.458618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09188 - 0.458618i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 + 0.947i)T \) |
| 5 | \( 1 + (0.681 + 2.12i)T \) |
| 19 | \( 1 + (2.31 + 3.69i)T \) |
| good | 3 | \( 1 + (-0.664 + 0.791i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.98 - 3.43i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.95 - 2.28i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.59 + 1.34i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.02 - 0.532i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (4.89 + 1.78i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.07 + 1.42i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.128 + 0.223i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 + (-3.49 + 4.16i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.50 - 1.27i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.32 + 7.54i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (5.94 + 2.16i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0505 - 0.286i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-12.9 - 4.72i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-6.61 + 1.16i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.32 - 1.57i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (4.91 - 5.85i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (1.37 + 1.15i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.10 - 5.37i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.410 - 0.489i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (3.18 - 18.0i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.46468972596987008797245793461, −10.24146474410430728803484828764, −9.181461668378975166695127604541, −8.408613191534339015207524293876, −8.100789981950537719369425915871, −6.77879564013069226191718707137, −5.19371763826708600865120717528, −4.09416003668781405992664015700, −2.39872976895748534282300377656, −1.42952082308796928444461893681,
1.28328274934858151588323876527, 3.53123920788402514187791120975, 4.34593570045907448065673723815, 6.15984277374449314770143458177, 6.78274977670941691318886279559, 7.84278538544719968793662939981, 8.542487977988967634647449094026, 9.726624391299077103487043237951, 10.34772276801396660659805893911, 11.16464256321992794692334832779
