| L(s) = 1 | + (4.08 + 5.62i)3-s + (−11.1 − 0.725i)5-s + 19.2i·7-s + (−6.59 + 20.2i)9-s + (13.7 + 42.4i)11-s + (−62.4 − 20.2i)13-s + (−41.5 − 65.7i)15-s + (16.7 − 23.0i)17-s + (76.5 + 55.6i)19-s + (−108. + 78.8i)21-s + (−112. + 36.5i)23-s + (123. + 16.1i)25-s + (37.4 − 12.1i)27-s + (−7.32 + 5.31i)29-s + (205. + 149. i)31-s + ⋯ |
| L(s) = 1 | + (0.786 + 1.08i)3-s + (−0.997 − 0.0648i)5-s + 1.04i·7-s + (−0.244 + 0.751i)9-s + (0.377 + 1.16i)11-s + (−1.33 − 0.432i)13-s + (−0.714 − 1.13i)15-s + (0.238 − 0.328i)17-s + (0.924 + 0.671i)19-s + (−1.12 + 0.819i)21-s + (−1.02 + 0.331i)23-s + (0.991 + 0.129i)25-s + (0.266 − 0.0866i)27-s + (−0.0468 + 0.0340i)29-s + (1.19 + 0.867i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.717294 + 1.29831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717294 + 1.29831i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (11.1 + 0.725i)T \) |
| good | 3 | \( 1 + (-4.08 - 5.62i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 - 19.2iT - 343T^{2} \) |
| 11 | \( 1 + (-13.7 - 42.4i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (62.4 + 20.2i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-16.7 + 23.0i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-76.5 - 55.6i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (112. - 36.5i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (7.32 - 5.31i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-205. - 149. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-261. - 85.0i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-89.4 + 275. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 371. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (94.9 + 130. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-94.9 - 130. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-60.1 + 185. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-132. - 408. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (549. - 756. i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-781. + 567. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-267. + 86.9i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (559. - 406. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (391. - 538. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (385. + 1.18e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-812. - 1.11e3i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−14.17939138850217891106838722200, −12.24185524079575694297363774943, −11.98363446460740890683432880844, −10.16548853464296386388714978777, −9.499948655225780374350563781922, −8.381385533200816653334947455364, −7.30766591664185023567372187102, −5.20512036010546915429661732519, −4.07779406704228772233956945308, −2.71375946417261820455378507318,
0.802031154567237478541520030380, 2.88616622253719976810750201798, 4.33379262800632616451124130299, 6.56413308856800564602359828018, 7.61963326488093812412213473558, 8.136256933119307913256635800606, 9.636351999334856739775349791507, 11.15933634079709360004637194522, 12.05055780172659422721538686246, 13.17220566105006119320309719144
