| L(s) = 1 | − 2.60i·5-s − 0.347i·7-s + 0.672·11-s − 2.08·13-s + 0.0255i·17-s + 5.29i·19-s + 8.89·23-s − 1.78·25-s + 0.154i·29-s − 7.87i·31-s − 0.906·35-s − 5.34·37-s + 6.80i·41-s − 11.2i·43-s + 2.56·47-s + ⋯ |
| L(s) = 1 | − 1.16i·5-s − 0.131i·7-s + 0.202·11-s − 0.576·13-s + 0.00618i·17-s + 1.21i·19-s + 1.85·23-s − 0.356·25-s + 0.0287i·29-s − 1.41i·31-s − 0.153·35-s − 0.879·37-s + 1.06i·41-s − 1.71i·43-s + 0.373·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780783337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780783337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.60iT - 5T^{2} \) |
| 7 | \( 1 + 0.347iT - 7T^{2} \) |
| 11 | \( 1 - 0.672T + 11T^{2} \) |
| 13 | \( 1 + 2.08T + 13T^{2} \) |
| 17 | \( 1 - 0.0255iT - 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 - 8.89T + 23T^{2} \) |
| 29 | \( 1 - 0.154iT - 29T^{2} \) |
| 31 | \( 1 + 7.87iT - 31T^{2} \) |
| 37 | \( 1 + 5.34T + 37T^{2} \) |
| 41 | \( 1 - 6.80iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 + 3.97iT - 53T^{2} \) |
| 59 | \( 1 - 6.14T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 5.55iT - 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 3.39iT - 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 10.6iT - 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−8.255726997236412279850319961516, −7.27517229703481411335786431404, −6.73680364060725562459425643091, −5.54690482100659380505546942700, −5.26438193482025947709806793424, −4.32745413215609847652630208177, −3.68294104832339790392076955714, −2.53633673156331646846508273474, −1.49596899290508220679563347576, −0.55735017397351443592953531511,
1.03838797769946445608995988657, 2.45075530477470851087272248649, 2.91048580127185535617607310336, 3.77603459830018999257767920407, 4.85938524833631748762354791884, 5.38594954970429580359061858374, 6.51178576491390185906273081908, 7.01146313001669696930998118701, 7.32963535263453167170353499748, 8.506905187261543269734725255289
