| L(s) = 1 | + (11.3 − 11.3i)2-s + (−244. − 101. i)3-s + 252. i·4-s + (230. − 556. i)5-s + (−3.93e3 + 1.62e3i)6-s + (406. + 981. i)7-s + (8.70e3 + 8.70e3i)8-s + (3.55e4 + 3.55e4i)9-s + (−3.70e3 − 8.95e3i)10-s + (−1.64e4 + 6.83e3i)11-s + (2.55e4 − 6.17e4i)12-s + 3.77e4i·13-s + (1.57e4 + 6.54e3i)14-s + (−1.12e5 + 1.12e5i)15-s + 6.88e4·16-s + (3.82e4 + 3.42e5i)17-s + ⋯ |
| L(s) = 1 | + (0.503 − 0.503i)2-s + (−1.74 − 0.721i)3-s + 0.493i·4-s + (0.164 − 0.397i)5-s + (−1.23 + 0.513i)6-s + (0.0639 + 0.154i)7-s + (0.751 + 0.751i)8-s + (1.80 + 1.80i)9-s + (−0.117 − 0.283i)10-s + (−0.339 + 0.140i)11-s + (0.356 − 0.859i)12-s + 0.366i·13-s + (0.109 + 0.0455i)14-s + (−0.574 + 0.574i)15-s + 0.262·16-s + (0.111 + 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.804825 + 0.410463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804825 + 0.410463i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-3.82e4 - 3.42e5i)T \) |
| good | 2 | \( 1 + (-11.3 + 11.3i)T - 512iT^{2} \) |
| 3 | \( 1 + (244. + 101. i)T + (1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-230. + 556. i)T + (-1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-406. - 981. i)T + (-2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (1.64e4 - 6.83e3i)T + (1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 3.77e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (1.21e5 - 1.21e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (2.32e6 - 9.65e5i)T + (1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-1.21e6 + 2.93e6i)T + (-1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-6.18e6 - 2.56e6i)T + (1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (1.25e7 + 5.20e6i)T + (9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-9.84e6 - 2.37e7i)T + (-2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-1.15e7 - 1.15e7i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 - 4.00e6iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (6.96e7 - 6.96e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (5.73e7 + 5.73e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (-3.33e7 - 8.05e7i)T + (-8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.58e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (2.24e8 + 9.31e7i)T + (3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-1.42e8 + 3.42e8i)T + (-4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (1.55e8 - 6.44e7i)T + (8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-3.88e8 + 3.88e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + 1.54e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (2.98e8 - 7.19e8i)T + (-5.37e17 - 5.37e17i)T^{2} \) |
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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
−17.13296302791831928926007249536, −16.08136954713743156411636120199, −13.56231731299101374525049230676, −12.50031129822410853862507608326, −11.80729119528876103094240358618, −10.53209745832854232757568349695, −7.80685054050968477461598312935, −6.03804366246904039957348977006, −4.57604928184952574651478075029, −1.66250072926081331856431997985,
0.48892702718737480788990845258, 4.49503138394798039662728931427, 5.67684614318556796160955415050, 6.78114232699368646938169525452, 10.02792520902196227726701045216, 10.76794921961943713466115157109, 12.26389053448711748829488993375, 14.07227490242427214734983341955, 15.58174062704935483107150350045, 16.23367863859724204602837325530
