| L(s) = 1 | + (−1.10 − 2.65i)2-s + (8.44 + 12.6i)3-s + (5.46 − 5.46i)4-s + (3.31 − 16.6i)5-s + (24.2 − 36.3i)6-s + (6.66 + 33.5i)7-s + (−63.0 − 26.1i)8-s + (−57.4 + 138. i)9-s + (−47.9 + 9.53i)10-s + (−39.6 − 26.4i)11-s + (115. + 22.9i)12-s + (−160. − 160. i)13-s + (81.7 − 54.6i)14-s + (238. − 98.9i)15-s + 72.5i·16-s + (−269. − 104. i)17-s + ⋯ |
| L(s) = 1 | + (−0.275 − 0.664i)2-s + (0.938 + 1.40i)3-s + (0.341 − 0.341i)4-s + (0.132 − 0.667i)5-s + (0.674 − 1.00i)6-s + (0.136 + 0.684i)7-s + (−0.985 − 0.408i)8-s + (−0.708 + 1.71i)9-s + (−0.479 + 0.0953i)10-s + (−0.327 − 0.218i)11-s + (0.800 + 0.159i)12-s + (−0.950 − 0.950i)13-s + (0.417 − 0.278i)14-s + (1.06 − 0.439i)15-s + 0.283i·16-s + (−0.932 − 0.360i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.39747 - 0.0985484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39747 - 0.0985484i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (269. + 104. i)T \) |
| good | 2 | \( 1 + (1.10 + 2.65i)T + (-11.3 + 11.3i)T^{2} \) |
| 3 | \( 1 + (-8.44 - 12.6i)T + (-30.9 + 74.8i)T^{2} \) |
| 5 | \( 1 + (-3.31 + 16.6i)T + (-577. - 239. i)T^{2} \) |
| 7 | \( 1 + (-6.66 - 33.5i)T + (-2.21e3 + 918. i)T^{2} \) |
| 11 | \( 1 + (39.6 + 26.4i)T + (5.60e3 + 1.35e4i)T^{2} \) |
| 13 | \( 1 + (160. + 160. i)T + 2.85e4iT^{2} \) |
| 19 | \( 1 + (-244. - 590. i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 + (-466. + 697. i)T + (-1.07e5 - 2.58e5i)T^{2} \) |
| 29 | \( 1 + (-506. - 100. i)T + (6.53e5 + 2.70e5i)T^{2} \) |
| 31 | \( 1 + (590. - 394. i)T + (3.53e5 - 8.53e5i)T^{2} \) |
| 37 | \( 1 + (-236. - 354. i)T + (-7.17e5 + 1.73e6i)T^{2} \) |
| 41 | \( 1 + (-102. - 515. i)T + (-2.61e6 + 1.08e6i)T^{2} \) |
| 43 | \( 1 + (203. - 491. i)T + (-2.41e6 - 2.41e6i)T^{2} \) |
| 47 | \( 1 + (-339. - 339. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (19.3 + 46.6i)T + (-5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 + (599. + 248. i)T + (8.56e6 + 8.56e6i)T^{2} \) |
| 61 | \( 1 + (-2.05e3 + 409. i)T + (1.27e7 - 5.29e6i)T^{2} \) |
| 67 | \( 1 + 7.93e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + (-873. - 1.30e3i)T + (-9.72e6 + 2.34e7i)T^{2} \) |
| 73 | \( 1 + (58.2 - 292. i)T + (-2.62e7 - 1.08e7i)T^{2} \) |
| 79 | \( 1 + (784. + 524. i)T + (1.49e7 + 3.59e7i)T^{2} \) |
| 83 | \( 1 + (8.27e3 - 3.42e3i)T + (3.35e7 - 3.35e7i)T^{2} \) |
| 89 | \( 1 + (5.06e3 - 5.06e3i)T - 6.27e7iT^{2} \) |
| 97 | \( 1 + (7.79e3 + 1.55e3i)T + (8.17e7 + 3.38e7i)T^{2} \) |
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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
−18.47265704266524400246651830291, −16.46996654487253830427802995981, −15.40879713947563543254055804029, −14.52579683744033990977561385862, −12.47332033535778189454719083741, −10.70879816722629088159398720249, −9.657173041090498208069199766127, −8.557129941165244026758752051053, −5.15454656756004045385444875394, −2.80731869665131171431943072474,
2.57703441334264101857921133413, 6.95338216765235339277451193838, 7.30941184879912154382674221964, 8.980016803281366068712049834249, 11.47490568607883622451115200342, 13.05994953606210214045316493093, 14.19645039620549885835172714609, 15.35000282130893075425247009662, 17.24331659927234969392544026148, 18.02668757106255313149836443009
