| L(s) = 1 | + 5-s + 3i·7-s − 2·11-s + (3 − 2i)13-s − 3·17-s − 6·23-s − 4·25-s + 6i·29-s + 3i·35-s − 3·37-s − 10i·41-s + 9i·43-s + 7i·47-s − 2·49-s − 6i·53-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13i·7-s − 0.603·11-s + (0.832 − 0.554i)13-s − 0.727·17-s − 1.25·23-s − 0.800·25-s + 1.11i·29-s + 0.507i·35-s − 0.493·37-s − 1.56i·41-s + 1.37i·43-s + 1.02i·47-s − 0.285·49-s − 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5484268275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5484268275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 9iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 4iT - 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−8.869512354462601399654551415056, −8.211010742775666279718371238507, −7.50517299867787095471162236233, −6.41158513193436131253351348246, −5.84981594954039124744514809038, −5.32218129745857483059681628483, −4.32113556113918426629979024348, −3.27840343935163274625854080462, −2.42769242749467014929904873173, −1.59996790756401227109921359506,
0.14748134850072264780085898210, 1.53107726336924127940583740583, 2.40029102597249136560902495368, 3.69055430505702448338018136457, 4.19373498968219773073118436251, 5.11433187728673941286343198260, 6.14448099354391109716302156388, 6.52346341183531242304633628487, 7.57128160868109870392226098742, 8.013296097028327413479950037862
