| L(s) = 1 | − 6i·5-s + 10i·13-s + 30·17-s − 11·25-s − 42i·29-s + 70i·37-s + 18·41-s + 49·49-s + 90i·53-s − 22i·61-s + 60·65-s + 110·73-s − 180i·85-s − 78·89-s + 130·97-s + ⋯ |
| L(s) = 1 | − 1.20i·5-s + 0.769i·13-s + 1.76·17-s − 0.440·25-s − 1.44i·29-s + 1.89i·37-s + 0.439·41-s + 0.999·49-s + 1.69i·53-s − 0.360i·61-s + 0.923·65-s + 1.50·73-s − 2.11i·85-s − 0.876·89-s + 1.34·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.197136586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197136586\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 6iT - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 30T + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 42iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 70iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 90iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 + 22iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + 78T + 7.92e3T^{2} \) |
| 97 | \( 1 - 130T + 9.40e3T^{2} \) |
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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
−8.672243604038588728806440295169, −8.069139900162541371249948811727, −7.33929360619155702067078727105, −6.25304481870702018068317624973, −5.53688525712605129901747656301, −4.71576452255682235439086051087, −4.01872993866539170731746451675, −2.89067897473477821940656745346, −1.59388882548360788565103950305, −0.74165003080220603859027193295,
0.865474931233982975697968176621, 2.26239435494238405967030413519, 3.22007957771164262530692609195, 3.73653459998382439837131896505, 5.17551353633541638097665801459, 5.74519862775647288290230797803, 6.69826706816541714831195323449, 7.39890861876936883290483201856, 7.948441057641080198572468752874, 8.953065711073489174126056811550
