L(s) = 1 | + (−0.764 + 1.18i)2-s + (2.31 − 0.910i)3-s + (−0.830 − 1.81i)4-s + (0.243 − 1.61i)5-s + (−0.691 + 3.45i)6-s + (2.38 + 1.14i)7-s + (2.79 + 0.403i)8-s + (2.35 − 2.18i)9-s + (1.73 + 1.52i)10-s + (−0.368 + 0.396i)11-s + (−3.58 − 3.46i)12-s + (−4.52 − 1.03i)13-s + (−3.18 + 1.95i)14-s + (−0.904 − 3.96i)15-s + (−2.62 + 3.02i)16-s + (0.0399 − 0.532i)17-s + ⋯ |
L(s) = 1 | + (−0.540 + 0.841i)2-s + (1.33 − 0.525i)3-s + (−0.415 − 0.909i)4-s + (0.108 − 0.721i)5-s + (−0.282 + 1.41i)6-s + (0.900 + 0.433i)7-s + (0.989 + 0.142i)8-s + (0.784 − 0.727i)9-s + (0.548 + 0.481i)10-s + (−0.110 + 0.119i)11-s + (−1.03 − 1.00i)12-s + (−1.25 − 0.286i)13-s + (−0.852 + 0.523i)14-s + (−0.233 − 1.02i)15-s + (−0.655 + 0.755i)16-s + (0.00968 − 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65401 - 0.0301936i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65401 - 0.0301936i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.764 - 1.18i)T \) |
| 7 | \( 1 + (-2.38 - 1.14i)T \) |
good | 3 | \( 1 + (-2.31 + 0.910i)T + (2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (-0.243 + 1.61i)T + (-4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (0.368 - 0.396i)T + (-0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (4.52 + 1.03i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.0399 + 0.532i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-7.33 + 4.23i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0838 + 1.11i)T + (-22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (1.09 + 2.27i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (3.77 - 6.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.60 + 3.81i)T + (-13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (3.12 - 3.91i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (0.436 - 0.348i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (9.59 - 2.96i)T + (38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (-5.76 - 8.45i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (1.18 + 7.85i)T + (-56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (6.16 - 9.04i)T + (-22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-8.61 - 4.97i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.21 + 1.54i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.26 - 0.388i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (4.77 + 8.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (12.4 - 2.84i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (11.6 - 10.7i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−11.24190748326490787813050953060, −9.829206027692567356842919130433, −9.144302420017732763782609264833, −8.502781351226126365035549597361, −7.64478656418833672062348017375, −7.13333963867047723834234865629, −5.40374129994020309530931069302, −4.74267114945141188778875041031, −2.73233200822753249534749523404, −1.39480865361711263406960597740,
1.88330256768582952412423891429, 2.98733074588860941174931942182, 3.85470736369414668349896040210, 5.04760411863048870024103724558, 7.23783221730446789352651920028, 7.79898982742103702575128564661, 8.667683126878906556017406248868, 9.792727187254989380317007235374, 10.02673158079365244911154369396, 11.19065746528322761408287636753