| L(s) = 1 | + (−0.210 + 0.507i)2-s + (−5.96 + 8.92i)3-s + (11.1 + 11.1i)4-s + (1.57 + 7.90i)5-s + (−3.27 − 4.90i)6-s + (7.85 − 39.4i)7-s + (−16.0 + 6.66i)8-s + (−13.0 − 31.5i)9-s + (−4.34 − 0.864i)10-s + (37.6 − 25.1i)11-s + (−165. + 32.8i)12-s + (199. − 199. i)13-s + (18.3 + 12.2i)14-s + (−79.9 − 33.1i)15-s + 241. i·16-s + (14.9 + 288. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0525 + 0.126i)2-s + (−0.662 + 0.991i)3-s + (0.693 + 0.693i)4-s + (0.0629 + 0.316i)5-s + (−0.0910 − 0.136i)6-s + (0.160 − 0.805i)7-s + (−0.251 + 0.104i)8-s + (−0.161 − 0.389i)9-s + (−0.0434 − 0.00864i)10-s + (0.310 − 0.207i)11-s + (−1.14 + 0.228i)12-s + (1.18 − 1.18i)13-s + (0.0938 + 0.0627i)14-s + (−0.355 − 0.147i)15-s + 0.943i·16-s + (0.0515 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.864661 + 0.724054i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.864661 + 0.724054i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-14.9 - 288. i)T \) |
| good | 2 | \( 1 + (0.210 - 0.507i)T + (-11.3 - 11.3i)T^{2} \) |
| 3 | \( 1 + (5.96 - 8.92i)T + (-30.9 - 74.8i)T^{2} \) |
| 5 | \( 1 + (-1.57 - 7.90i)T + (-577. + 239. i)T^{2} \) |
| 7 | \( 1 + (-7.85 + 39.4i)T + (-2.21e3 - 918. i)T^{2} \) |
| 11 | \( 1 + (-37.6 + 25.1i)T + (5.60e3 - 1.35e4i)T^{2} \) |
| 13 | \( 1 + (-199. + 199. i)T - 2.85e4iT^{2} \) |
| 19 | \( 1 + (33.4 - 80.7i)T + (-9.21e4 - 9.21e4i)T^{2} \) |
| 23 | \( 1 + (447. + 670. i)T + (-1.07e5 + 2.58e5i)T^{2} \) |
| 29 | \( 1 + (511. - 101. i)T + (6.53e5 - 2.70e5i)T^{2} \) |
| 31 | \( 1 + (365. + 244. i)T + (3.53e5 + 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-971. + 1.45e3i)T + (-7.17e5 - 1.73e6i)T^{2} \) |
| 41 | \( 1 + (384. - 1.93e3i)T + (-2.61e6 - 1.08e6i)T^{2} \) |
| 43 | \( 1 + (142. + 343. i)T + (-2.41e6 + 2.41e6i)T^{2} \) |
| 47 | \( 1 + (2.08e3 - 2.08e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-1.95e3 + 4.71e3i)T + (-5.57e6 - 5.57e6i)T^{2} \) |
| 59 | \( 1 + (-1.09e3 + 451. i)T + (8.56e6 - 8.56e6i)T^{2} \) |
| 61 | \( 1 + (4.82e3 + 960. i)T + (1.27e7 + 5.29e6i)T^{2} \) |
| 67 | \( 1 - 2.20e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-3.39e3 + 5.08e3i)T + (-9.72e6 - 2.34e7i)T^{2} \) |
| 73 | \( 1 + (-657. - 3.30e3i)T + (-2.62e7 + 1.08e7i)T^{2} \) |
| 79 | \( 1 + (1.35e3 - 907. i)T + (1.49e7 - 3.59e7i)T^{2} \) |
| 83 | \( 1 + (6.47e3 + 2.68e3i)T + (3.35e7 + 3.35e7i)T^{2} \) |
| 89 | \( 1 + (-7.23e3 - 7.23e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (1.01e4 - 2.01e3i)T + (8.17e7 - 3.38e7i)T^{2} \) |
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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
−18.03407647448951297959660233054, −16.80411780653167638244854293074, −16.14191421303036546674559411243, −14.82883831857759436308069143491, −12.88728463714001124160640827979, −11.10283740276908390343593259596, −10.45835151401296427603075647975, −8.109900973524730581819052900137, −6.18348326298667641521923507622, −3.84384147155320120898195108210,
1.55759425157495319355541174201, 5.72387353417653154228088922372, 6.96440465280964712979156109391, 9.243217961743878600525264803560, 11.35538785874160531687693811990, 12.02778919913260077396153362724, 13.67860635205091417709338495111, 15.33594699922673858190604986485, 16.60372663316617228252846632366, 18.23914497889427809597843223323
