| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−2.78 − 8.55i)3-s + (4.94 + 15.2i)4-s + (−55.6 − 4.77i)5-s + (−11.1 + 34.2i)6-s + 235.·7-s + (19.7 − 60.8i)8-s + (−65.5 + 47.6i)9-s + (169. + 146. i)10-s + (−470. − 341. i)11-s + (116. − 84.6i)12-s + (−547. + 397. i)13-s + (−762. − 554. i)14-s + (113. + 490. i)15-s + (−207. + 150. i)16-s + (79.1 − 243. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.996 − 0.0854i)5-s + (−0.126 + 0.388i)6-s + 1.81·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.534 + 0.463i)10-s + (−1.17 − 0.851i)11-s + (0.233 − 0.169i)12-s + (−0.898 + 0.652i)13-s + (−1.04 − 0.755i)14-s + (0.130 + 0.562i)15-s + (−0.202 + 0.146i)16-s + (0.0664 − 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8927234845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8927234845\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 3 | \( 1 + (2.78 + 8.55i)T \) |
| 5 | \( 1 + (55.6 + 4.77i)T \) |
| good | 7 | \( 1 - 235.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (470. + 341. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (547. - 397. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-79.1 + 243. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (884. - 2.72e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.48e3 - 1.07e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-334. - 1.02e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-946. + 2.91e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-6.69e3 + 4.86e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.46e3 + 4.69e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 4.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.45e3 - 1.06e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-9.35e3 - 2.87e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.42e4 + 1.03e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.86e4 - 2.81e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (8.42e3 - 2.59e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.18e4 - 6.72e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (6.70e4 + 4.87e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.20e4 - 3.71e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-361. + 1.11e3i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (6.22e3 + 4.52e3i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.57e4 - 1.40e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.85797416342234860634386206409, −11.28406149149689596365620834863, −10.45652265464903912518349075061, −8.717515684046886743843005618765, −7.913268649351808693055391582129, −7.43657746317231051458680645301, −5.47595041768233487162842566042, −4.21970565013759191840422055622, −2.41783110325270638167905664560, −1.03545772529588430609623539509,
0.45917923228675116356723175162, 2.47196652829764635174752912463, 4.70027546533333904105264747164, 5.01142401300834411231709638802, 7.09612520788205341658520253150, 7.915079437165659569290744845423, 8.657615117822623967844923332801, 10.16374407688672410846997339263, 10.95701312882222901967921311480, 11.67311895335447657002164141269
