| L(s) = 1 | + (−0.718 − 0.829i)2-s + (0.397 − 2.76i)4-s + (−4.86 − 2.21i)5-s + (2.07 − 7.07i)7-s + (−6.27 + 4.03i)8-s + (1.65 + 5.62i)10-s + (10.0 + 8.71i)11-s + (−17.3 + 5.08i)13-s + (−7.36 + 3.36i)14-s + (−2.87 − 0.844i)16-s + (−30.2 + 4.35i)17-s + (29.7 + 4.28i)19-s + (−8.07 + 12.5i)20-s − 14.6i·22-s + (−15.5 − 16.9i)23-s + ⋯ |
| L(s) = 1 | + (−0.359 − 0.414i)2-s + (0.0994 − 0.691i)4-s + (−0.972 − 0.443i)5-s + (0.296 − 1.01i)7-s + (−0.784 + 0.503i)8-s + (0.165 + 0.562i)10-s + (0.914 + 0.792i)11-s + (−1.33 + 0.390i)13-s + (−0.526 + 0.240i)14-s + (−0.179 − 0.0527i)16-s + (−1.78 + 0.256i)17-s + (1.56 + 0.225i)19-s + (−0.403 + 0.628i)20-s − 0.663i·22-s + (−0.675 − 0.736i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0891380 + 0.469672i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0891380 + 0.469672i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + (15.5 + 16.9i)T \) |
| good | 2 | \( 1 + (0.718 + 0.829i)T + (-0.569 + 3.95i)T^{2} \) |
| 5 | \( 1 + (4.86 + 2.21i)T + (16.3 + 18.8i)T^{2} \) |
| 7 | \( 1 + (-2.07 + 7.07i)T + (-41.2 - 26.4i)T^{2} \) |
| 11 | \( 1 + (-10.0 - 8.71i)T + (17.2 + 119. i)T^{2} \) |
| 13 | \( 1 + (17.3 - 5.08i)T + (142. - 91.3i)T^{2} \) |
| 17 | \( 1 + (30.2 - 4.35i)T + (277. - 81.4i)T^{2} \) |
| 19 | \( 1 + (-29.7 - 4.28i)T + (346. + 101. i)T^{2} \) |
| 29 | \( 1 + (4.53 + 31.5i)T + (-806. + 236. i)T^{2} \) |
| 31 | \( 1 + (11.7 - 7.54i)T + (399. - 874. i)T^{2} \) |
| 37 | \( 1 + (60.2 - 27.5i)T + (896. - 1.03e3i)T^{2} \) |
| 41 | \( 1 + (3.19 - 6.99i)T + (-1.10e3 - 1.27e3i)T^{2} \) |
| 43 | \( 1 + (-19.3 + 30.1i)T + (-768. - 1.68e3i)T^{2} \) |
| 47 | \( 1 - 39.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.4 + 93.6i)T + (-2.36e3 - 1.51e3i)T^{2} \) |
| 59 | \( 1 + (-12.8 + 3.77i)T + (2.92e3 - 1.88e3i)T^{2} \) |
| 61 | \( 1 + (3.69 + 5.75i)T + (-1.54e3 + 3.38e3i)T^{2} \) |
| 67 | \( 1 + (1.90 - 1.64i)T + (638. - 4.44e3i)T^{2} \) |
| 71 | \( 1 + (65.1 + 75.1i)T + (-717. + 4.98e3i)T^{2} \) |
| 73 | \( 1 + (-8.23 + 57.2i)T + (-5.11e3 - 1.50e3i)T^{2} \) |
| 79 | \( 1 + (14.5 + 49.3i)T + (-5.25e3 + 3.37e3i)T^{2} \) |
| 83 | \( 1 + (-51.3 + 23.4i)T + (4.51e3 - 5.20e3i)T^{2} \) |
| 89 | \( 1 + (-39.9 + 62.2i)T + (-3.29e3 - 7.20e3i)T^{2} \) |
| 97 | \( 1 + (-1.71 - 0.785i)T + (6.16e3 + 7.11e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.80802132610597545665508176308, −10.60026407744162340696163945293, −9.757048794080689944579183244235, −8.823049149572919561393315065663, −7.52683745884867726609377020260, −6.67961620175742606435325760241, −4.89381393095472343755716445680, −4.06694689385242163801725774419, −1.93884216566126453860422947752, −0.29132156919781628906063522136,
2.71937983678882884015282859310, 3.88628716595054206713044723753, 5.50241673714940305518206208609, 6.97653257494497674351097194645, 7.56900309720407049362591909333, 8.745990202275298011983364522048, 9.316475845734057384290439788400, 11.08731511426902045858107542219, 11.82227901902609790525417429091, 12.30517856764915819891132899256
