L(s) = 1 | − 0.134·2-s + (−2.15 + 2.15i)3-s − 1.98·4-s + (1.82 + 1.29i)5-s + (0.290 − 0.290i)6-s + 1.90i·7-s + 0.536·8-s − 6.29i·9-s + (−0.245 − 0.173i)10-s + (−0.290 − 0.290i)11-s + (4.27 − 4.27i)12-s + (3.60 − 0.173i)13-s − 0.257i·14-s + (−6.71 + 1.15i)15-s + 3.89·16-s + (−2.53 + 2.53i)17-s + ⋯ |
L(s) = 1 | − 0.0951·2-s + (−1.24 + 1.24i)3-s − 0.990·4-s + (0.816 + 0.576i)5-s + (0.118 − 0.118i)6-s + 0.721i·7-s + 0.189·8-s − 2.09i·9-s + (−0.0777 − 0.0549i)10-s + (−0.0875 − 0.0875i)11-s + (1.23 − 1.23i)12-s + (0.998 − 0.0481i)13-s − 0.0687i·14-s + (−1.73 + 0.298i)15-s + 0.972·16-s + (−0.615 + 0.615i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.332904 + 0.442454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.332904 + 0.442454i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.82 - 1.29i)T \) |
| 13 | \( 1 + (-3.60 + 0.173i)T \) |
good | 2 | \( 1 + 0.134T + 2T^{2} \) |
| 3 | \( 1 + (2.15 - 2.15i)T - 3iT^{2} \) |
| 7 | \( 1 - 1.90iT - 7T^{2} \) |
| 11 | \( 1 + (0.290 + 0.290i)T + 11iT^{2} \) |
| 17 | \( 1 + (2.53 - 2.53i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3.15 - 3.15i)T + 19iT^{2} \) |
| 23 | \( 1 + (2.27 + 2.27i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.40iT - 29T^{2} \) |
| 31 | \( 1 + (-2.02 + 2.02i)T - 31iT^{2} \) |
| 37 | \( 1 + 5.32iT - 37T^{2} \) |
| 41 | \( 1 + (1.51 - 1.51i)T - 41iT^{2} \) |
| 43 | \( 1 + (-0.888 - 0.888i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.94iT - 47T^{2} \) |
| 53 | \( 1 + (1.09 - 1.09i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.31 + 8.31i)T - 59iT^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 + 0.939T + 67T^{2} \) |
| 71 | \( 1 + (7.37 - 7.37i)T - 71iT^{2} \) |
| 73 | \( 1 + 6.63T + 73T^{2} \) |
| 79 | \( 1 - 4.39iT - 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 + (10.0 - 10.0i)T - 89iT^{2} \) |
| 97 | \( 1 + 4.39T + 97T^{2} \) |
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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
−15.33900117668743132902221213371, −14.28359693793779263197893418039, −13.00407925198356294859840901796, −11.62700329579126878421940178074, −10.50142200532292456655426604940, −9.777926995103257129221057698187, −8.694235655361333921412445203602, −6.13441084724006496807754619305, −5.41496974358630568270895729773, −3.90845422570094098027227911901,
1.08644087346231578302571292467, 4.79025585203389526205423182940, 5.90696980874424973850839559141, 7.21935838656273575774134866803, 8.677998386283046688142644055334, 10.09985069496473112829791244146, 11.35196522802303792416459528823, 12.59110053615028309535190585306, 13.56517512055824109478521904972, 13.73198301886497196750279466565