| L(s) = 1 | + (1.23 − 0.509i)2-s + (−2.63 − 0.523i)3-s + (−1.57 + 1.57i)4-s + (0.902 + 1.35i)5-s + (−3.50 + 0.696i)6-s + (4.92 − 7.37i)7-s + (−3.17 + 7.66i)8-s + (−1.66 − 0.690i)9-s + (1.79 + 1.20i)10-s + (−1.61 − 8.12i)11-s + (4.96 − 3.31i)12-s + (16.1 + 16.1i)13-s + (2.30 − 11.5i)14-s + (−1.66 − 4.02i)15-s + 2.13i·16-s + (−15.7 + 6.50i)17-s + ⋯ |
| L(s) = 1 | + (0.615 − 0.254i)2-s + (−0.876 − 0.174i)3-s + (−0.393 + 0.393i)4-s + (0.180 + 0.270i)5-s + (−0.584 + 0.116i)6-s + (0.703 − 1.05i)7-s + (−0.396 + 0.957i)8-s + (−0.185 − 0.0767i)9-s + (0.179 + 0.120i)10-s + (−0.146 − 0.738i)11-s + (0.413 − 0.276i)12-s + (1.24 + 1.24i)13-s + (0.164 − 0.827i)14-s + (−0.111 − 0.268i)15-s + 0.133i·16-s + (−0.923 + 0.382i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.826822 - 0.0856580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.826822 - 0.0856580i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (15.7 - 6.50i)T \) |
| good | 2 | \( 1 + (-1.23 + 0.509i)T + (2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (2.63 + 0.523i)T + (8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-0.902 - 1.35i)T + (-9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-4.92 + 7.37i)T + (-18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (1.61 + 8.12i)T + (-111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (-16.1 - 16.1i)T + 169iT^{2} \) |
| 19 | \( 1 + (1.20 - 0.500i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (26.2 - 5.21i)T + (488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-13.7 + 9.19i)T + (321. - 776. i)T^{2} \) |
| 31 | \( 1 + (-2.68 + 13.4i)T + (-887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-18.9 - 3.77i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (14.6 - 21.9i)T + (-643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (21.8 + 9.06i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-26.8 - 26.8i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-26.1 + 10.8i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-9.12 + 22.0i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (42.9 + 28.6i)T + (1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 70.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (102. + 20.4i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-19.7 - 29.5i)T + (-2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (1.41 + 7.11i)T + (-5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-51.5 - 124. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-38.6 + 38.6i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-65.9 + 44.0i)T + (3.60e3 - 8.69e3i)T^{2} \) |
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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
−18.34314955642293031537716558318, −17.46103281469524957169471418125, −16.42211126425268658752897902344, −14.20658895060597692219260157906, −13.47540807565330781227131655767, −11.72924110954539054577060807973, −10.91494465694063083659657437898, −8.447955698882083847016988963644, −6.23435970428835425516639192668, −4.23580729155528584116831458285,
4.96829166293346856188793079113, 5.96378396233134840155527313587, 8.725023094536014144876395223523, 10.58150786388914135967592008732, 12.09363895728878166008830294004, 13.42743120668347517943644738544, 14.97817464415429984493199258463, 15.90890456212894979788277660554, 17.75386148768329824702562750019, 18.27491347558555853869415128540
