L(s) = 1 | + (0.300 − 1.38i)2-s + (−0.987 + 0.156i)3-s + (−1.81 − 0.830i)4-s + (2.22 + 0.222i)5-s + (−0.0804 + 1.41i)6-s + (3.70 + 3.70i)7-s + (−1.69 + 2.26i)8-s + (0.951 − 0.309i)9-s + (0.976 − 3.00i)10-s + (−0.352 − 0.114i)11-s + (1.92 + 0.535i)12-s + (2.36 − 4.64i)13-s + (6.22 − 4.00i)14-s + (−2.23 + 0.128i)15-s + (2.62 + 3.02i)16-s + (−0.155 + 0.980i)17-s + ⋯ |
L(s) = 1 | + (0.212 − 0.977i)2-s + (−0.570 + 0.0903i)3-s + (−0.909 − 0.415i)4-s + (0.995 + 0.0995i)5-s + (−0.0328 + 0.576i)6-s + (1.39 + 1.39i)7-s + (−0.598 + 0.800i)8-s + (0.317 − 0.103i)9-s + (0.308 − 0.951i)10-s + (−0.106 − 0.0344i)11-s + (0.556 + 0.154i)12-s + (0.655 − 1.28i)13-s + (1.66 − 1.07i)14-s + (−0.576 + 0.0330i)15-s + (0.655 + 0.755i)16-s + (−0.0376 + 0.237i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31929 - 0.595055i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31929 - 0.595055i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.300 + 1.38i)T \) |
| 3 | \( 1 + (0.987 - 0.156i)T \) |
| 5 | \( 1 + (-2.22 - 0.222i)T \) |
good | 7 | \( 1 + (-3.70 - 3.70i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.352 + 0.114i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.36 + 4.64i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.155 - 0.980i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (2.30 - 1.67i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.21 + 6.30i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-0.959 + 1.32i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.91 - 5.39i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (3.78 + 1.92i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-0.959 - 2.95i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-0.302 + 0.302i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.817 + 5.16i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (0.685 + 4.32i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (1.32 + 4.07i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.66 - 8.20i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (4.13 + 0.654i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (3.73 - 5.14i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (12.0 - 6.12i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (7.17 + 5.21i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.07 + 6.78i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (12.1 + 3.93i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 1.72i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.63149458211146907777728929536, −10.63469598069082461303509235535, −10.20109197821596109177531290646, −8.775637227924613254931583671889, −8.299754505393630057218267942874, −6.08950224652104785119559694807, −5.51774854737775525190407774537, −4.60621419741414961580153417864, −2.74026994121004545261467682972, −1.59883520215873560931594326768,
1.46578295708219410109913006231, 4.13529757965155936681691318662, 4.86071977269459146685385571083, 5.99042082784170046888031739769, 6.92121000235424483682572616991, 7.77741788919048438122771278433, 8.896491894790311105349553885737, 9.948873542153235600124872903635, 10.94080584523376976126834797266, 11.82718892740137724471797163101