| L(s) = 1 | + (1.40 − 0.131i)2-s + (0.740 − 0.671i)3-s + (1.96 − 0.371i)4-s + (−0.117 − 0.469i)5-s + (0.954 − 1.04i)6-s + (−3.34 − 2.74i)7-s + (2.71 − 0.782i)8-s + (0.0980 − 0.995i)9-s + (−0.227 − 0.645i)10-s + (−2.39 + 1.13i)11-s + (1.20 − 1.59i)12-s + (2.00 − 1.20i)13-s + (−5.06 − 3.41i)14-s + (−0.402 − 0.268i)15-s + (3.72 − 1.46i)16-s + (4.73 − 3.16i)17-s + ⋯ |
| L(s) = 1 | + (0.995 − 0.0933i)2-s + (0.427 − 0.387i)3-s + (0.982 − 0.185i)4-s + (−0.0525 − 0.209i)5-s + (0.389 − 0.425i)6-s + (−1.26 − 1.03i)7-s + (0.960 − 0.276i)8-s + (0.0326 − 0.331i)9-s + (−0.0719 − 0.204i)10-s + (−0.721 + 0.341i)11-s + (0.348 − 0.460i)12-s + (0.555 − 0.333i)13-s + (−1.35 − 0.913i)14-s + (−0.103 − 0.0693i)15-s + (0.930 − 0.365i)16-s + (1.14 − 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.23412 - 1.85843i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.23412 - 1.85843i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 + 0.131i)T \) |
| 3 | \( 1 + (-0.740 + 0.671i)T \) |
| good | 5 | \( 1 + (0.117 + 0.469i)T + (-4.40 + 2.35i)T^{2} \) |
| 7 | \( 1 + (3.34 + 2.74i)T + (1.36 + 6.86i)T^{2} \) |
| 11 | \( 1 + (2.39 - 1.13i)T + (6.97 - 8.50i)T^{2} \) |
| 13 | \( 1 + (-2.00 + 1.20i)T + (6.12 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-4.73 + 3.16i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 + 1.51i)T + (-5.51 - 18.1i)T^{2} \) |
| 23 | \( 1 + (2.37 - 0.721i)T + (19.1 - 12.7i)T^{2} \) |
| 29 | \( 1 + (2.94 - 1.05i)T + (22.4 - 18.3i)T^{2} \) |
| 31 | \( 1 + (-2.57 - 6.21i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.0938 - 0.632i)T + (-35.4 - 10.7i)T^{2} \) |
| 41 | \( 1 + (2.19 + 1.17i)T + (22.7 + 34.0i)T^{2} \) |
| 43 | \( 1 + (3.93 - 4.34i)T + (-4.21 - 42.7i)T^{2} \) |
| 47 | \( 1 + (2.44 - 12.3i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-6.37 - 2.27i)T + (40.9 + 33.6i)T^{2} \) |
| 59 | \( 1 + (-11.7 - 7.03i)T + (27.8 + 52.0i)T^{2} \) |
| 61 | \( 1 + (9.96 + 0.489i)T + (60.7 + 5.97i)T^{2} \) |
| 67 | \( 1 + (-0.377 + 7.68i)T + (-66.6 - 6.56i)T^{2} \) |
| 71 | \( 1 + (-0.818 + 0.0805i)T + (69.6 - 13.8i)T^{2} \) |
| 73 | \( 1 + (-1.18 + 0.971i)T + (14.2 - 71.5i)T^{2} \) |
| 79 | \( 1 + (-0.918 + 0.182i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-1.80 - 12.1i)T + (-79.4 + 24.0i)T^{2} \) |
| 89 | \( 1 + (-2.74 - 0.831i)T + (74.0 + 49.4i)T^{2} \) |
| 97 | \( 1 + (0.944 - 0.391i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.21449909967929305236151918491, −9.539521507062770802436954961452, −8.127322036005455225089998537746, −7.30234727076503237103072040798, −6.68921058714169267675621214767, −5.66471206753615369722961963116, −4.58594698125797223481473893387, −3.42549284052072388057455612011, −2.85816277272965575830804199984, −1.07319492556098121765434531846,
2.19801775979384310979422367457, 3.23879332479831203450656179299, 3.78941632206014045242241864009, 5.31947918583623568199685806761, 5.89035250532359845468047799399, 6.78103635380796665684092374373, 7.928591921824350100739848359966, 8.713448093530644058178000360295, 9.876542109617071247285332238317, 10.41914312021531256589517630289
