| L(s) = 1 | + (1.85 − 5.09i)2-s + (7.50 − 1.32i)3-s + (−10.2 − 8.58i)4-s + (−16.5 + 13.8i)5-s + (7.17 − 40.6i)6-s + (−1.44 + 2.49i)7-s + (12.4 − 7.16i)8-s + (−21.4 + 7.81i)9-s + (40.0 + 109. i)10-s + (68.6 + 118. i)11-s + (−88.2 − 50.9i)12-s + (87.9 + 15.5i)13-s + (10.0 + 11.9i)14-s + (−105. + 126. i)15-s + (−50.5 − 286. i)16-s + (−493. − 179. i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 1.27i)2-s + (0.834 − 0.147i)3-s + (−0.639 − 0.536i)4-s + (−0.661 + 0.554i)5-s + (0.199 − 1.13i)6-s + (−0.0294 + 0.0509i)7-s + (0.193 − 0.111i)8-s + (−0.265 + 0.0964i)9-s + (0.400 + 1.09i)10-s + (0.567 + 0.983i)11-s + (−0.612 − 0.353i)12-s + (0.520 + 0.0917i)13-s + (0.0512 + 0.0610i)14-s + (−0.470 + 0.560i)15-s + (−0.197 − 1.12i)16-s + (−1.70 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.34792 - 1.10802i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34792 - 1.10802i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-357. + 50.5i)T \) |
| good | 2 | \( 1 + (-1.85 + 5.09i)T + (-12.2 - 10.2i)T^{2} \) |
| 3 | \( 1 + (-7.50 + 1.32i)T + (76.1 - 27.7i)T^{2} \) |
| 5 | \( 1 + (16.5 - 13.8i)T + (108. - 615. i)T^{2} \) |
| 7 | \( 1 + (1.44 - 2.49i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-68.6 - 118. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-87.9 - 15.5i)T + (2.68e4 + 9.76e3i)T^{2} \) |
| 17 | \( 1 + (493. + 179. i)T + (6.39e4 + 5.36e4i)T^{2} \) |
| 23 | \( 1 + (479. + 401. i)T + (4.85e4 + 2.75e5i)T^{2} \) |
| 29 | \( 1 + (8.32 + 22.8i)T + (-5.41e5 + 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-112. - 65.0i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + 621. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.47e3 + 436. i)T + (2.65e6 - 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-1.88e3 + 1.57e3i)T + (5.93e5 - 3.36e6i)T^{2} \) |
| 47 | \( 1 + (1.52e3 - 556. i)T + (3.73e6 - 3.13e6i)T^{2} \) |
| 53 | \( 1 + (-2.18e3 + 2.59e3i)T + (-1.37e6 - 7.77e6i)T^{2} \) |
| 59 | \( 1 + (1.20e3 - 3.30e3i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-1.72e3 - 1.44e3i)T + (2.40e6 + 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-890. - 2.44e3i)T + (-1.54e7 + 1.29e7i)T^{2} \) |
| 71 | \( 1 + (6.07e3 + 7.23e3i)T + (-4.41e6 + 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-463. - 2.62e3i)T + (-2.66e7 + 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-1.94e3 + 342. i)T + (3.66e7 - 1.33e7i)T^{2} \) |
| 83 | \( 1 + (3.44e3 - 5.96e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-8.06e3 - 1.42e3i)T + (5.89e7 + 2.14e7i)T^{2} \) |
| 97 | \( 1 + (-2.39e3 + 6.59e3i)T + (-6.78e7 - 5.69e7i)T^{2} \) |
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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
−17.97844420447393451882684110243, −15.85524888773645084386414582605, −14.45262696329245060254425801927, −13.40781756511339822642007213946, −11.95189849542176840541741196355, −10.94137918131800043139354175296, −9.216871014678406267997858107712, −7.31790746017581964491748770553, −4.06202622091989974104034878444, −2.45460348152822752547204420606,
4.01764326387887060817530992484, 6.08289670985384580584826247782, 7.957969701098592813426430867313, 8.878160436143700903999042959132, 11.39399287721591823827203251654, 13.38629161235375279078828338399, 14.28116564971873636770690645001, 15.52998348765483808948947404585, 16.23272067656771609372961881632, 17.61316787145730950693010102714
