| L(s) = 1 | + (1.30 + 1.14i)3-s + (−0.761 + 0.639i)5-s + (1.35 − 0.492i)7-s + (0.396 + 2.97i)9-s + (−2.56 − 2.15i)11-s + (−0.337 − 1.91i)13-s + (−1.72 − 0.0362i)15-s + (1.16 + 2.01i)17-s + (3.38 − 5.86i)19-s + (2.32 + 0.901i)21-s + (−8.41 − 3.06i)23-s + (−0.696 + 3.95i)25-s + (−2.87 + 4.32i)27-s + (0.847 − 4.80i)29-s + (−5.81 − 2.11i)31-s + ⋯ |
| L(s) = 1 | + (0.752 + 0.658i)3-s + (−0.340 + 0.285i)5-s + (0.511 − 0.186i)7-s + (0.132 + 0.991i)9-s + (−0.773 − 0.648i)11-s + (−0.0936 − 0.531i)13-s + (−0.444 − 0.00936i)15-s + (0.281 + 0.487i)17-s + (0.776 − 1.34i)19-s + (0.507 + 0.196i)21-s + (−1.75 − 0.638i)23-s + (−0.139 + 0.790i)25-s + (−0.553 + 0.832i)27-s + (0.157 − 0.892i)29-s + (−1.04 − 0.380i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16294 + 0.344978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16294 + 0.344978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 - 1.14i)T \) |
| good | 5 | \( 1 + (0.761 - 0.639i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 0.492i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.56 + 2.15i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.337 + 1.91i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.16 - 2.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.38 + 5.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.41 + 3.06i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.847 + 4.80i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (5.81 + 2.11i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.0829 + 0.143i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.87 - 10.6i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.82 - 5.73i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-6.14 + 2.23i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + (-3.02 + 2.53i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (7.91 - 2.88i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.00383 + 0.0217i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.53 - 7.85i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.26 - 3.92i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.00 - 5.70i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.898 + 5.09i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-1.91 + 3.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 + 4.53i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.91316647814853541167106279489, −13.03294517667520252198758855801, −11.43929052171744359803271737782, −10.63109700699153775137892383425, −9.568085123575681398692937804473, −8.240226442706818056956493053688, −7.57468562286328620561759570510, −5.58760639510411974810927286354, −4.17572146636358562765252027995, −2.75974797778797820020452839801,
2.01684233813988193756287545892, 3.87948875983784716088081480211, 5.57317879583325711418212384971, 7.30976276931837044449447368007, 7.983423096356040017579562531040, 9.140105418181650584314450461679, 10.31404919118047683982170828863, 12.04402383381191244047776887125, 12.33253453010258350738249318867, 13.83258901621926210898624899569
