| L(s) = 1 | + 1.66i·2-s − 2.79·3-s + 1.21·4-s − 4.66i·6-s − 6.72i·7-s + 8.70i·8-s − 1.18·9-s + (−6.62 − 8.78i)11-s − 3.40·12-s − 2.91i·13-s + 11.2·14-s − 9.63·16-s − 30.9i·17-s − 1.98i·18-s + 4.49i·19-s + ⋯ |
| L(s) = 1 | + 0.833i·2-s − 0.931·3-s + 0.304·4-s − 0.776i·6-s − 0.960i·7-s + 1.08i·8-s − 0.131·9-s + (−0.601 − 0.798i)11-s − 0.283·12-s − 0.224i·13-s + 0.800·14-s − 0.602·16-s − 1.82i·17-s − 0.110i·18-s + 0.236i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.761664 - 0.379683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.761664 - 0.379683i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (6.62 + 8.78i)T \) |
| good | 2 | \( 1 - 1.66iT - 4T^{2} \) |
| 3 | \( 1 + 2.79T + 9T^{2} \) |
| 7 | \( 1 + 6.72iT - 49T^{2} \) |
| 13 | \( 1 + 2.91iT - 169T^{2} \) |
| 17 | \( 1 + 30.9iT - 289T^{2} \) |
| 19 | \( 1 - 4.49iT - 361T^{2} \) |
| 23 | \( 1 + 13.0T + 529T^{2} \) |
| 29 | \( 1 + 44.8iT - 841T^{2} \) |
| 31 | \( 1 - 26.1T + 961T^{2} \) |
| 37 | \( 1 - 28.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 24.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.28iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.33T + 2.20e3T^{2} \) |
| 53 | \( 1 + 93.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 108.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 54.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 9.01T + 4.48e3T^{2} \) |
| 71 | \( 1 + 1.24T + 5.04e3T^{2} \) |
| 73 | \( 1 - 24.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 108. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 8.27iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 84.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 160.T + 9.40e3T^{2} \) |
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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
−11.35119179035044036753737624897, −10.87100021050466173373667820625, −9.745794989308527562791885818183, −8.217070500476121413279164778422, −7.49526935159263561072020925873, −6.44547509160254358990041545073, −5.68795839024220649962936597016, −4.68791557915054827516217297358, −2.82747079177328037792819457459, −0.46145367619635709143238452964,
1.69502287434634950291595396452, 2.96250637443861823513601532310, 4.57182529911746539481806034457, 5.84461809727691939216184834178, 6.55909016290439144040710173788, 7.972440944671738289025492431476, 9.202860963412672378351949937093, 10.36616417913330261578689287078, 10.87334877412813624194903593359, 11.87814300253995420930901858267
