L(s) = 1 | − i·2-s − 4-s + (0.485 − 0.841i)5-s + (−2.47 − 0.940i)7-s + i·8-s + (−0.841 − 0.485i)10-s + (−2.28 − 1.32i)11-s + (−2.23 − 2.83i)13-s + (−0.940 + 2.47i)14-s + 16-s + 0.627·17-s + (0.415 − 0.239i)19-s + (−0.485 + 0.841i)20-s + (−1.32 + 2.28i)22-s + 3.48i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.217 − 0.376i)5-s + (−0.934 − 0.355i)7-s + 0.353i·8-s + (−0.266 − 0.153i)10-s + (−0.689 − 0.398i)11-s + (−0.619 − 0.785i)13-s + (−0.251 + 0.660i)14-s + 0.250·16-s + 0.152·17-s + (0.0952 − 0.0550i)19-s + (−0.108 + 0.188i)20-s + (−0.281 + 0.487i)22-s + 0.727i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0138 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1137701242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1137701242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.47 + 0.940i)T \) |
| 13 | \( 1 + (2.23 + 2.83i)T \) |
good | 5 | \( 1 + (-0.485 + 0.841i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.28 + 1.32i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 0.627T + 17T^{2} \) |
| 19 | \( 1 + (-0.415 + 0.239i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3.48iT - 23T^{2} \) |
| 29 | \( 1 + (4.77 - 2.75i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.59 + 0.920i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.51T + 37T^{2} \) |
| 41 | \( 1 + (-0.880 - 1.52i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.424 - 0.734i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.32 - 7.49i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.50 - 4.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 5.11T + 59T^{2} \) |
| 61 | \( 1 + (11.0 - 6.37i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.40 - 4.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.14 + 5.28i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.8 + 6.83i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.04 + 1.80i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 8.12T + 89T^{2} \) |
| 97 | \( 1 + (-2.72 - 1.57i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.479598618873572609932849527309, −9.209362843848786165315753459809, −7.948937189434197772476188291688, −7.42193035481392821087477315953, −6.16403449256932172564167078426, −5.42101012711871152667957787594, −4.56023011846112468172838239124, −3.34435545825892196266987904628, −2.80851953876540858286444290535, −1.31253405325242242582878038763,
0.04418063177742970791487921254, 2.16374997367262269874734215794, 3.11734990449771286917717456367, 4.31056270645676956808305691230, 5.16635225469864645887480750170, 6.14000998262612866292779901816, 6.69853002300918193369201935456, 7.45852959123407108552463506532, 8.316575140279138555849677932189, 9.235592393300205119039148887825