| L(s) = 1 | + (−1.36 − 0.377i)2-s + (−1.15 + 1.29i)3-s + (1.71 + 1.02i)4-s + (−0.216 − 0.161i)5-s + (2.05 − 1.32i)6-s + (1.25 − 1.91i)7-s + (−1.94 − 2.05i)8-s + (−0.347 − 2.97i)9-s + (0.234 + 0.301i)10-s + (−1.15 + 0.135i)11-s + (−3.30 + 1.03i)12-s + (0.114 − 0.384i)13-s + (−2.43 + 2.13i)14-s + (0.458 − 0.0946i)15-s + (1.88 + 3.52i)16-s + (1.67 − 0.295i)17-s + ⋯ |
| L(s) = 1 | + (−0.963 − 0.266i)2-s + (−0.664 + 0.746i)3-s + (0.857 + 0.514i)4-s + (−0.0969 − 0.0722i)5-s + (0.840 − 0.542i)6-s + (0.475 − 0.723i)7-s + (−0.688 − 0.724i)8-s + (−0.115 − 0.993i)9-s + (0.0742 + 0.0954i)10-s + (−0.349 + 0.0408i)11-s + (−0.954 + 0.298i)12-s + (0.0318 − 0.106i)13-s + (−0.651 + 0.569i)14-s + (0.118 − 0.0244i)15-s + (0.470 + 0.882i)16-s + (0.405 − 0.0715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.699849 - 0.115594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.699849 - 0.115594i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.36 + 0.377i)T \) |
| 3 | \( 1 + (1.15 - 1.29i)T \) |
| good | 5 | \( 1 + (0.216 + 0.161i)T + (1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.25 + 1.91i)T + (-2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (1.15 - 0.135i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.114 + 0.384i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 0.295i)T + (15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-7.02 - 1.23i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-3.41 + 2.24i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (-0.517 - 0.488i)T + (1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-3.99 + 0.232i)T + (30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-1.53 - 0.559i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-1.65 - 6.97i)T + (-36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.122 - 0.0527i)T + (29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.629 + 10.8i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-5.73 + 3.30i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.97 - 0.932i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-1.74 + 0.877i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (7.12 - 6.71i)T + (3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (6.80 - 5.70i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (11.6 + 9.76i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.61 + 6.81i)T + (-70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (11.7 + 2.79i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (-5.88 + 7.00i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.86 + 5.19i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−11.51638475384286562302404549284, −10.32345468035925335714756361266, −10.08493831686587424230579977511, −8.899385383088357417968259494600, −7.86479454445821730023632847842, −6.93147040769276617011772237613, −5.66308074792825392490609298688, −4.39318047147863723649158859603, −3.09398985802450458388530453922, −0.922227051465824270227555397734,
1.25771881530669615198038262321, 2.72564995462262833349175421549, 5.21737681448239665715175427328, 5.82314056987810685875460339691, 7.13810029033538602053922211624, 7.70562761870812919783088689216, 8.737548589548055344310645652574, 9.740704618133652513112755381743, 10.86640367517254679853110799197, 11.58059711239567518813715241813
