| L(s) = 1 | + (−1.31 + 4.04i)2-s + (−8.13 − 5.91i)4-s + (8.20 − 7.59i)5-s − 28.2·7-s + (7.06 − 5.13i)8-s + (19.8 + 43.1i)10-s + (−15.8 + 48.6i)11-s + (−7.92 − 24.4i)13-s + (37.1 − 114. i)14-s + (−13.3 − 41.2i)16-s + (75.4 − 54.8i)17-s + (95.3 − 69.3i)19-s + (−111. + 13.2i)20-s + (−175. − 127. i)22-s + (29.1 − 89.8i)23-s + ⋯ |
| L(s) = 1 | + (−0.464 + 1.42i)2-s + (−1.01 − 0.738i)4-s + (0.734 − 0.678i)5-s − 1.52·7-s + (0.312 − 0.226i)8-s + (0.629 + 1.36i)10-s + (−0.433 + 1.33i)11-s + (−0.169 − 0.520i)13-s + (0.709 − 2.18i)14-s + (−0.209 − 0.643i)16-s + (1.07 − 0.782i)17-s + (1.15 − 0.836i)19-s + (−1.24 + 0.148i)20-s + (−1.70 − 1.23i)22-s + (0.264 − 0.814i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.822487 - 0.0580450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.822487 - 0.0580450i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-8.20 + 7.59i)T \) |
| good | 2 | \( 1 + (1.31 - 4.04i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 + 28.2T + 343T^{2} \) |
| 11 | \( 1 + (15.8 - 48.6i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (7.92 + 24.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-75.4 + 54.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-95.3 + 69.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-29.1 + 89.8i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-46.7 - 33.9i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-32.7 + 23.8i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (126. + 389. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (31.4 + 96.7i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 1.56T + 7.95e4T^{2} \) |
| 47 | \( 1 + (265. + 193. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (368. + 267. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-120. - 369. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (149. - 459. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (122. - 88.9i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-158. - 115. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-186. + 573. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (484. + 352. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (654. - 475. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-201. + 620. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (1.25e3 + 914. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−12.15482285450151196690052652669, −10.10034588183329715451664469168, −9.645710507436475671214603181457, −8.861288721322451644556873054590, −7.50434139411791047076187106186, −6.85065664567930431496366882522, −5.70337267847874164701689595806, −4.93968129290218529667304406496, −2.81627101635240223933718527222, −0.42025949913479772133867328321,
1.34710502665443188401882610130, 3.10857174033140144407917009485, 3.32824959799487650240609729574, 5.73235635997645725337560240269, 6.57519184159950653231760745072, 8.178907248409274433674844971331, 9.581011249117904306917202141831, 9.826460063366013048272379611713, 10.72662295312182078657357056107, 11.66333250548851819144242894614
