| L(s) = 1 | + (−5.57 − 5.57i)2-s + (−9.69 + 4.01i)3-s + 30.2i·4-s + (21.4 + 51.6i)5-s + (76.4 + 31.6i)6-s + (−20.6 + 49.8i)7-s + (−9.76 + 9.76i)8-s + (−93.9 + 93.9i)9-s + (168. − 407. i)10-s + (−383. − 158. i)11-s + (−121. − 293. i)12-s + 14.1i·13-s + (393. − 162. i)14-s + (−415. − 415. i)15-s + 1.07e3·16-s + (−829. + 855. i)17-s + ⋯ |
| L(s) = 1 | + (−0.986 − 0.986i)2-s + (−0.621 + 0.257i)3-s + 0.945i·4-s + (0.383 + 0.924i)5-s + (0.867 + 0.359i)6-s + (−0.159 + 0.384i)7-s + (−0.0539 + 0.0539i)8-s + (−0.386 + 0.386i)9-s + (0.534 − 1.28i)10-s + (−0.956 − 0.396i)11-s + (−0.243 − 0.587i)12-s + 0.0232i·13-s + (0.536 − 0.222i)14-s + (−0.476 − 0.476i)15-s + 1.05·16-s + (−0.696 + 0.717i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0311 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.248063 + 0.240448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248063 + 0.240448i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (829. - 855. i)T \) |
| good | 2 | \( 1 + (5.57 + 5.57i)T + 32iT^{2} \) |
| 3 | \( 1 + (9.69 - 4.01i)T + (171. - 171. i)T^{2} \) |
| 5 | \( 1 + (-21.4 - 51.6i)T + (-2.20e3 + 2.20e3i)T^{2} \) |
| 7 | \( 1 + (20.6 - 49.8i)T + (-1.18e4 - 1.18e4i)T^{2} \) |
| 11 | \( 1 + (383. + 158. i)T + (1.13e5 + 1.13e5i)T^{2} \) |
| 13 | \( 1 - 14.1iT - 3.71e5T^{2} \) |
| 19 | \( 1 + (-960. - 960. i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + (3.74e3 + 1.55e3i)T + (4.55e6 + 4.55e6i)T^{2} \) |
| 29 | \( 1 + (-1.42e3 - 3.43e3i)T + (-1.45e7 + 1.45e7i)T^{2} \) |
| 31 | \( 1 + (-5.24e3 + 2.17e3i)T + (2.02e7 - 2.02e7i)T^{2} \) |
| 37 | \( 1 + (1.36e4 - 5.63e3i)T + (4.90e7 - 4.90e7i)T^{2} \) |
| 41 | \( 1 + (-1.70e3 + 4.10e3i)T + (-8.19e7 - 8.19e7i)T^{2} \) |
| 43 | \( 1 + (-3.71e3 + 3.71e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 - 2.46e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.93e4 - 1.93e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (-2.47e4 + 2.47e4i)T - 7.14e8iT^{2} \) |
| 61 | \( 1 + (334. - 808. i)T + (-5.97e8 - 5.97e8i)T^{2} \) |
| 67 | \( 1 + 2.36e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.16e4 - 4.81e3i)T + (1.27e9 - 1.27e9i)T^{2} \) |
| 73 | \( 1 + (-1.58e4 - 3.83e4i)T + (-1.46e9 + 1.46e9i)T^{2} \) |
| 79 | \( 1 + (2.59e3 + 1.07e3i)T + (2.17e9 + 2.17e9i)T^{2} \) |
| 83 | \( 1 + (7.33e4 + 7.33e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.13e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (6.55e3 + 1.58e4i)T + (-6.07e9 + 6.07e9i)T^{2} \) |
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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
−18.29405163197579459153521804397, −17.38857104885253383065017692459, −15.86585852882133056100268273854, −14.09760794022231124600749148143, −12.11989710745935593235850156584, −10.78892419115186661219368080164, −10.20537791850866220538759035694, −8.355836756988115710508427211630, −5.89619677302723928054631295529, −2.56763644249481876393330751615,
0.38883250591623774363644344491, 5.47079664870664624002961310440, 7.06081739302861782827229204974, 8.646488839734116487134983668195, 9.969724138809643902750992319128, 12.02199064117744408646592431329, 13.50363969499548846469269117649, 15.53979721343657492912727968622, 16.46338891429037738424060829951, 17.60944128017020284734812861215
