| L(s) = 1 | + (−1.15 + 2.79i)2-s + (−0.675 − 0.451i)3-s + (−3.65 − 3.65i)4-s + (7.87 − 1.56i)5-s + (2.04 − 1.36i)6-s + (−7.70 − 1.53i)7-s + (3.25 − 1.34i)8-s + (−3.19 − 7.70i)9-s + (−4.74 + 23.8i)10-s + (4.48 + 6.71i)11-s + (0.818 + 4.11i)12-s + (−0.798 + 0.798i)13-s + (13.2 − 19.7i)14-s + (−6.02 − 2.49i)15-s − 9.98i·16-s + (−6.50 + 15.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 + 1.39i)2-s + (−0.225 − 0.150i)3-s + (−0.913 − 0.913i)4-s + (1.57 − 0.313i)5-s + (0.340 − 0.227i)6-s + (−1.10 − 0.218i)7-s + (0.407 − 0.168i)8-s + (−0.354 − 0.856i)9-s + (−0.474 + 2.38i)10-s + (0.407 + 0.610i)11-s + (0.0682 + 0.342i)12-s + (−0.0614 + 0.0614i)13-s + (0.943 − 1.41i)14-s + (−0.401 − 0.166i)15-s − 0.624i·16-s + (−0.382 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.511999 + 0.401606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.511999 + 0.401606i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (6.50 - 15.7i)T \) |
| good | 2 | \( 1 + (1.15 - 2.79i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (0.675 + 0.451i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-7.87 + 1.56i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (7.70 + 1.53i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-4.48 - 6.71i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (0.798 - 0.798i)T - 169iT^{2} \) |
| 19 | \( 1 + (-1.07 + 2.59i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (7.13 - 4.76i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (0.599 + 3.01i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (7.13 - 10.6i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-19.6 - 13.1i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-21.4 - 4.26i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (8.89 + 21.4i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-55.6 + 55.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (22.9 - 55.5i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (25.3 - 10.5i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-7.11 + 35.7i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-88.1 - 58.8i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (59.8 - 11.9i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-52.9 - 79.2i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (109. + 45.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (61.4 + 61.4i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-4.79 - 24.1i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−18.34185721371765964283059242195, −17.34923114309340131539487218247, −16.84059516519479983649285171802, −15.32265673835626759469710137124, −13.99815504523868948258784051924, −12.61465096353372725939527281782, −9.867551564307791548475527173092, −9.018357298548798655347051572956, −6.72462311994906145545123176733, −5.88255570330040301972242750514,
2.61910761517579352530306967518, 6.04506245827218849988613335155, 9.163530198287405734721457993134, 10.05265098610722854820701312546, 11.21415092652783609050225251999, 12.89744688834960769133397107806, 13.95421080932840473954065299354, 16.33206356284071898279452875555, 17.57780915739113958991639636417, 18.66263507539011643057641889251
