| L(s) = 1 | + (−3.30 − 2.25i)2-s + (4.44 − 7.82i)3-s + (5.79 + 14.9i)4-s + (−2.53 + 1.46i)5-s + (−32.3 + 15.7i)6-s + (−39.8 − 23.0i)7-s + (14.5 − 62.3i)8-s + (−41.4 − 69.6i)9-s + (11.6 + 0.896i)10-s + (27.1 − 46.9i)11-s + (142. + 21.0i)12-s + (−174. + 100. i)13-s + (79.5 + 165. i)14-s + (0.170 + 26.3i)15-s + (−188. + 172. i)16-s − 88.7·17-s + ⋯ |
| L(s) = 1 | + (−0.825 − 0.564i)2-s + (0.494 − 0.869i)3-s + (0.362 + 0.932i)4-s + (−0.101 + 0.0586i)5-s + (−0.898 + 0.438i)6-s + (−0.813 − 0.469i)7-s + (0.227 − 0.973i)8-s + (−0.511 − 0.859i)9-s + (0.116 + 0.00896i)10-s + (0.224 − 0.388i)11-s + (0.989 + 0.146i)12-s + (−1.03 + 0.596i)13-s + (0.405 + 0.846i)14-s + (0.000758 + 0.117i)15-s + (−0.737 + 0.675i)16-s − 0.306·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0993094 + 0.501181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0993094 + 0.501181i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.30 + 2.25i)T \) |
| 3 | \( 1 + (-4.44 + 7.82i)T \) |
| good | 5 | \( 1 + (2.53 - 1.46i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (39.8 + 23.0i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-27.1 + 46.9i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (174. - 100. i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 88.7T + 8.35e4T^{2} \) |
| 19 | \( 1 + 357.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-214. + 123. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-18.7 - 10.8i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-349. + 201. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.26e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.10e3 + 1.91e3i)T + (-1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-678. + 1.17e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-3.12e3 - 1.80e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 4.12e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (2.64e3 + 4.57e3i)T + (-6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (4.77e3 + 2.75e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.09e3 - 5.35e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 4.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 9.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (1.09e3 + 635. i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.26e3 + 2.19e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 5.90e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (928. - 1.60e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−13.01215667344183263855701562930, −12.23639244594403175389952368194, −11.04219114573874288984487732330, −9.672480853132975978593222245174, −8.750120570067924368392547167638, −7.42716575323922001532028298193, −6.58386411533717648002855747467, −3.69899731378170031217993165832, −2.20506361839364110087169081950, −0.30638900935447191618245710313,
2.60814172326957933144916997843, 4.72888810703110056546126304870, 6.24047642893998451122004939190, 7.76188750566123064653898798411, 8.931476883852904094940446201140, 9.784757360391124011688343230188, 10.64514868895051938198853061529, 12.19927788146875389881877766601, 13.75688351483183153508264478301, 15.12389910097979033451229385984
