| L(s) = 1 | + (1.06 + 0.936i)2-s + (0.595 − 0.803i)3-s + (0.247 + 1.98i)4-s + (−0.820 + 0.905i)5-s + (1.38 − 0.293i)6-s + (1.57 + 2.94i)7-s + (−1.59 + 2.33i)8-s + (−0.290 − 0.956i)9-s + (−1.71 + 0.191i)10-s + (−0.303 − 1.21i)11-s + (1.74 + 0.983i)12-s + (−0.108 + 2.21i)13-s + (−1.08 + 4.58i)14-s + (0.238 + 1.19i)15-s + (−3.87 + 0.981i)16-s + (−0.0354 + 0.178i)17-s + ⋯ |
| L(s) = 1 | + (0.749 + 0.661i)2-s + (0.343 − 0.463i)3-s + (0.123 + 0.992i)4-s + (−0.367 + 0.404i)5-s + (0.564 − 0.119i)6-s + (0.594 + 1.11i)7-s + (−0.564 + 0.825i)8-s + (−0.0967 − 0.318i)9-s + (−0.543 + 0.0605i)10-s + (−0.0914 − 0.365i)11-s + (0.502 + 0.283i)12-s + (−0.0302 + 0.615i)13-s + (−0.290 + 1.22i)14-s + (0.0615 + 0.309i)15-s + (−0.969 + 0.245i)16-s + (−0.00860 + 0.0432i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37898 + 1.91750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37898 + 1.91750i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.06 - 0.936i)T \) |
| 3 | \( 1 + (-0.595 + 0.803i)T \) |
| good | 5 | \( 1 + (0.820 - 0.905i)T + (-0.490 - 4.97i)T^{2} \) |
| 7 | \( 1 + (-1.57 - 2.94i)T + (-3.88 + 5.82i)T^{2} \) |
| 11 | \( 1 + (0.303 + 1.21i)T + (-9.70 + 5.18i)T^{2} \) |
| 13 | \( 1 + (0.108 - 2.21i)T + (-12.9 - 1.27i)T^{2} \) |
| 17 | \( 1 + (0.0354 - 0.178i)T + (-15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (1.41 - 0.507i)T + (14.6 - 12.0i)T^{2} \) |
| 23 | \( 1 + (0.213 + 0.259i)T + (-4.48 + 22.5i)T^{2} \) |
| 29 | \( 1 + (-0.318 - 0.531i)T + (-13.6 + 25.5i)T^{2} \) |
| 31 | \( 1 + (-6.89 + 2.85i)T + (21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 2.88i)T + (-23.4 + 28.6i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 6.33i)T + (-40.2 - 7.99i)T^{2} \) |
| 43 | \( 1 + (1.82 - 1.35i)T + (12.4 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.49 - 6.72i)T + (-17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-4.97 + 8.30i)T + (-24.9 - 46.7i)T^{2} \) |
| 59 | \( 1 + (0.0209 + 0.426i)T + (-58.7 + 5.78i)T^{2} \) |
| 61 | \( 1 + (5.50 - 0.816i)T + (58.3 - 17.7i)T^{2} \) |
| 67 | \( 1 + (-0.188 - 1.27i)T + (-64.1 + 19.4i)T^{2} \) |
| 71 | \( 1 + (-8.24 - 2.50i)T + (59.0 + 39.4i)T^{2} \) |
| 73 | \( 1 + (-2.76 + 5.17i)T + (-40.5 - 60.6i)T^{2} \) |
| 79 | \( 1 + (-7.20 - 4.81i)T + (30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.663 - 1.40i)T + (-52.6 - 64.1i)T^{2} \) |
| 89 | \( 1 + (-9.24 + 11.2i)T + (-17.3 - 87.2i)T^{2} \) |
| 97 | \( 1 + (-0.144 - 0.348i)T + (-68.5 + 68.5i)T^{2} \) |
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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
−10.91643917067252625792301611678, −9.381975684134224152579021144463, −8.530380764021703030787767455301, −7.982292454500876615636980856539, −7.03128456343097614164722763914, −6.21087862461838789269799794879, −5.34143612457674755970195413194, −4.25531030502437419046061804689, −3.09912773566942003774203128549, −2.12754224673587658362466733204,
0.941110277078337734964627521641, 2.48279608090814642062676852025, 3.73080792845022060771754569775, 4.46594537315351081530624881677, 5.11122630684380926014231918967, 6.42179860518756344592782088016, 7.54485437896984239516568231624, 8.380235602501524702071882287604, 9.463027484659158506139712292267, 10.41487688056479816078775812440
