| L(s) = 1 | + (2.55 − 1.85i)2-s + (−0.476 − 1.46i)3-s + (0.612 − 1.88i)4-s + (13.9 + 10.1i)5-s + (−3.93 − 2.86i)6-s + (−2.95 + 9.09i)7-s + (5.87 + 18.0i)8-s + (19.9 − 14.4i)9-s + 54.5·10-s − 3.05·12-s + (5.24 − 3.80i)13-s + (9.33 + 28.7i)14-s + (8.22 − 25.3i)15-s + (61.4 + 44.6i)16-s + (−76.4 − 55.5i)17-s + (24.0 − 73.9i)18-s + ⋯ |
| L(s) = 1 | + (0.903 − 0.656i)2-s + (−0.0916 − 0.282i)3-s + (0.0765 − 0.235i)4-s + (1.24 + 0.907i)5-s + (−0.268 − 0.194i)6-s + (−0.159 + 0.491i)7-s + (0.259 + 0.799i)8-s + (0.737 − 0.536i)9-s + 1.72·10-s − 0.0734·12-s + (0.111 − 0.0812i)13-s + (0.178 + 0.548i)14-s + (0.141 − 0.435i)15-s + (0.959 + 0.697i)16-s + (−1.09 − 0.792i)17-s + (0.314 − 0.968i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.91798 - 0.358697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.91798 - 0.358697i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + (-2.55 + 1.85i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (0.476 + 1.46i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-13.9 - 10.1i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (2.95 - 9.09i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-5.24 + 3.80i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (76.4 + 55.5i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (16.2 + 50.0i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 82.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-46.3 + 142. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (106. - 77.2i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (37.9 - 116. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-58.7 - 180. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 320.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (136. + 419. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-99.9 + 72.6i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (22.4 - 68.9i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (201. + 146. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 128.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (803. + 583. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-159. + 491. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (464. - 337. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (152. + 110. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 42.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.04e3 + 755. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.25997913720449108305150804454, −12.02712521883325162358125193728, −11.08984898830792984833503576292, −10.02723585077216470562717184218, −8.916721181963347497966088449693, −7.06128869032566631877834151129, −6.11724685282724011773052948737, −4.77657782213890731223878181158, −3.09356574066558705635638004487, −2.02581864090216019338956921803,
1.54739223235146594539040409140, 4.11679621855142179068906669392, 5.05235413173978154828714234748, 6.03516946383808583143849267322, 7.15523423946147743357859070998, 8.845923222827719468647013018850, 9.924775088369586611601023303124, 10.70640844431432347593514939722, 12.72560968488373897288380374506, 13.12009635159135744184202822582
