| L(s) = 1 | + (1.24 − 0.671i)2-s + (−0.339 − 1.69i)3-s + (1.09 − 1.67i)4-s + (2.10 + 0.759i)5-s + (−1.56 − 1.88i)6-s − 0.738i·7-s + (0.244 − 2.81i)8-s + (−2.76 + 1.15i)9-s + (3.12 − 0.467i)10-s + (−1.63 + 1.18i)11-s + (−3.21 − 1.29i)12-s + (3.83 + 2.78i)13-s + (−0.496 − 0.919i)14-s + (0.574 − 3.83i)15-s + (−1.58 − 3.67i)16-s + (−3.35 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.880 − 0.474i)2-s + (−0.196 − 0.980i)3-s + (0.549 − 0.835i)4-s + (0.940 + 0.339i)5-s + (−0.638 − 0.769i)6-s − 0.279i·7-s + (0.0863 − 0.996i)8-s + (−0.923 + 0.384i)9-s + (0.989 − 0.147i)10-s + (−0.491 + 0.357i)11-s + (−0.927 − 0.374i)12-s + (1.06 + 0.773i)13-s + (−0.132 − 0.245i)14-s + (0.148 − 0.988i)15-s + (−0.397 − 0.917i)16-s + (−0.813 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00842 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00842 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53227 - 1.54524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53227 - 1.54524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.671i)T \) |
| 3 | \( 1 + (0.339 + 1.69i)T \) |
| 5 | \( 1 + (-2.10 - 0.759i)T \) |
| good | 7 | \( 1 + 0.738iT - 7T^{2} \) |
| 11 | \( 1 + (1.63 - 1.18i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.83 - 2.78i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (3.35 - 1.08i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.82 - 0.919i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 0.736i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (7.24 + 2.35i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.78 + 2.20i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.994 + 0.722i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.02 - 1.40i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + (3.13 - 9.65i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-11.6 - 3.80i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.97 + 2.88i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.09 + 3.70i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-8.81 + 2.86i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (4.32 - 13.3i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (4.33 - 3.14i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (14.7 + 4.79i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.885 + 2.72i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (7.36 + 10.1i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.66 + 8.19i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.46648445430250210977533667784, −10.92907941351706008522564323192, −9.932537494854582851838304499018, −8.686465746543603770065505108131, −7.19870905926358002409692628334, −6.37510840913439939833928174309, −5.70215985742012596814317036519, −4.27839987174814066418378132354, −2.62116481595945091731760777629, −1.60607645968976556523048958037,
2.58866319656563290060974032905, 3.86520950810232097671242736254, 5.14734456308689625734190345298, 5.69401855216671253833913124204, 6.68224239898207975714524726792, 8.402404244818238592730909346698, 8.935499961186408998778609535538, 10.33310376567270796332013477261, 10.99719902685529423363347210495, 12.03538757712295367368168401628
