| L(s) = 1 | + (0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (2.17 − 0.506i)5-s + (−0.309 + 0.951i)6-s − 2.31·7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (2.05 + 0.870i)10-s + (−2.77 − 2.01i)11-s + (−0.809 + 0.587i)12-s + (3.98 − 2.89i)13-s + (−1.86 − 1.35i)14-s + (1.15 + 1.91i)15-s + (−0.809 + 0.587i)16-s + (−2.36 + 7.29i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.973 − 0.226i)5-s + (−0.126 + 0.388i)6-s − 0.873·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.651 + 0.275i)10-s + (−0.836 − 0.607i)11-s + (−0.233 + 0.169i)12-s + (1.10 − 0.803i)13-s + (−0.499 − 0.362i)14-s + (0.298 + 0.494i)15-s + (−0.202 + 0.146i)16-s + (−0.574 + 1.76i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42216 + 0.740344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42216 + 0.740344i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-2.17 + 0.506i)T \) |
| good | 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 + (2.77 + 2.01i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.98 + 2.89i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.36 - 7.29i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1 + 3.07i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.61 + 2.62i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.13 + 6.56i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 3.37i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.20 - 2.32i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.132 + 0.0964i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 6.71T + 43T^{2} \) |
| 47 | \( 1 + (-1.92 - 5.93i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.69 - 11.3i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.62 - 6.26i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.249 - 0.181i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (1.42 - 4.39i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-0.117 - 0.362i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0608 - 0.0442i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.72 - 14.5i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.403 + 1.24i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.58 + 2.60i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (5.31 + 16.3i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.28092308461185613219878890310, −12.66906948019869678846209142256, −10.93566118005413828046244701476, −10.21648272276676334531789943509, −8.975850420138978963962986721864, −8.057675825684046333610144888687, −6.18735424333493060798999412893, −5.77741574399437854828444367026, −4.14198077624082652590633160123, −2.75874616185726000106929453455,
1.99240304180795183403286585906, 3.32682123668003054015689684564, 5.18756501823203964174365978041, 6.33911216518976143659001103226, 7.21840416858112163442932910279, 9.024234598634644245274655809722, 9.842747843412162036450805519288, 10.91635917048554370609795707288, 12.09123303792780870064958755906, 13.04765598506603660387063155612
