L(s) = 1 | − 3i·3-s − 16.7i·5-s + 7·7-s − 9·9-s − 16.7i·11-s − 86.0i·13-s − 50.3·15-s + 87.5·17-s − 104. i·19-s − 21i·21-s − 171.·23-s − 157.·25-s + 27i·27-s − 88.8i·29-s + 53.1·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.50i·5-s + 0.377·7-s − 0.333·9-s − 0.459i·11-s − 1.83i·13-s − 0.867·15-s + 1.24·17-s − 1.26i·19-s − 0.218i·21-s − 1.55·23-s − 1.25·25-s + 0.192i·27-s − 0.568i·29-s + 0.307·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.976353844\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976353844\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 - 7T \) |
good | 5 | \( 1 + 16.7iT - 125T^{2} \) |
| 11 | \( 1 + 16.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 86.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 87.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 88.8iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 53.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 132. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 325.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 674. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 254. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 815. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 616. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 41.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 7.03T + 3.89e5T^{2} \) |
| 79 | \( 1 - 418.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 93.4iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 88.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 379.T + 9.12e5T^{2} \) |
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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
−8.498992962075921748906373038252, −8.127701412328784787071897773184, −7.48160243262038614285328438454, −6.05894833949761311484539635706, −5.47686008955671908676842844667, −4.74671034557739378104089223426, −3.54436152634923980173277832038, −2.35544105385072326868142228155, −0.989284910623583746390554197857, −0.53358382772039845266429721168,
1.66622182071861445474727056809, 2.62382270371013045931181497352, 3.78915500740565523165290574088, 4.29958502212448073093221341065, 5.70444071223968629661504281827, 6.31645633842506528337550377136, 7.33646434899791568135747827282, 7.84718997362870338403802272490, 9.079099114364069525145618269514, 9.803907417539335969485148393189