| L(s) = 1 | + 43.6·5-s − 50.6·7-s + 31.7·11-s + 150. i·13-s − 148.·17-s + (130. − 336. i)19-s − 135.·23-s + 1.28e3·25-s + 1.23e3i·29-s + 1.67e3i·31-s − 2.20e3·35-s + 109. i·37-s + 887. i·41-s + 695.·43-s + 1.41e3·47-s + ⋯ |
| L(s) = 1 | + 1.74·5-s − 1.03·7-s + 0.262·11-s + 0.891i·13-s − 0.514·17-s + (0.362 − 0.932i)19-s − 0.256·23-s + 2.04·25-s + 1.47i·29-s + 1.74i·31-s − 1.80·35-s + 0.0796i·37-s + 0.528i·41-s + 0.376·43-s + 0.638·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.356694466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356694466\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-130. + 336. i)T \) |
| good | 5 | \( 1 - 43.6T + 625T^{2} \) |
| 7 | \( 1 + 50.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 31.7T + 1.46e4T^{2} \) |
| 13 | \( 1 - 150. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 148.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 135.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.23e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.67e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 109. iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 887. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 695.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.41e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 1.17e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.30e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.46e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.10e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 2.10e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.13e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.19e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 7.73e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.82e3iT - 8.85e7T^{2} \) |
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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
−9.983289222499439137425602617794, −9.165861393275589086013278439442, −8.830966009801729151605574380378, −6.93395361848671413129901231999, −6.64486668185217308001133133301, −5.66018674654523363051249274797, −4.74501082619395078094299845705, −3.27171996415191740028862426130, −2.28175884163397297587010292383, −1.20137505486128704833716585395,
0.55949305087446512364315939750, 1.96376826815040508091766930762, 2.84133311014301972835196837746, 4.10711963696283943864736961933, 5.67015833006336921926721364500, 5.91397662529356171546412400820, 6.85443206716927611184611938032, 8.047385827812232490340151601956, 9.171616068534549447518868275505, 9.854270230120687601384557629828
