| L(s) = 1 | + (1.33 + 0.472i)2-s + (−0.891 − 0.453i)3-s + (1.55 + 1.25i)4-s + (1.01 − 1.99i)5-s + (−0.973 − 1.02i)6-s + (0.189 + 0.189i)7-s + (1.47 + 2.41i)8-s + (0.587 + 0.809i)9-s + (2.29 − 2.17i)10-s + (2.68 − 3.69i)11-s + (−0.812 − 1.82i)12-s + (−1.66 − 0.263i)13-s + (0.163 + 0.342i)14-s + (−1.80 + 1.31i)15-s + (0.828 + 3.91i)16-s + (2.52 + 4.95i)17-s + ⋯ |
| L(s) = 1 | + (0.942 + 0.333i)2-s + (−0.514 − 0.262i)3-s + (0.776 + 0.629i)4-s + (0.452 − 0.891i)5-s + (−0.397 − 0.418i)6-s + (0.0717 + 0.0717i)7-s + (0.522 + 0.852i)8-s + (0.195 + 0.269i)9-s + (0.724 − 0.689i)10-s + (0.808 − 1.11i)11-s + (−0.234 − 0.527i)12-s + (−0.462 − 0.0731i)13-s + (0.0436 + 0.0915i)14-s + (−0.466 + 0.340i)15-s + (0.207 + 0.978i)16-s + (0.611 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09375 - 0.0192113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09375 - 0.0192113i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 - 0.472i)T \) |
| 3 | \( 1 + (0.891 + 0.453i)T \) |
| 5 | \( 1 + (-1.01 + 1.99i)T \) |
| good | 7 | \( 1 + (-0.189 - 0.189i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.68 + 3.69i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.66 + 0.263i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.52 - 4.95i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.186 - 0.574i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.67 - 0.581i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.14 + 1.02i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.47 + 2.10i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.372 - 2.35i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (9.30 - 6.76i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.00980 + 0.00980i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.17 - 2.31i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (4.53 - 8.89i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-3.15 + 2.29i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.40 + 6.83i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (6.49 - 3.30i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-12.6 + 4.09i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.770 + 4.86i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-4.17 - 12.8i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (3.42 + 6.72i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-4.20 + 5.78i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.917 + 0.467i)T + (57.0 + 78.4i)T^{2} \) |
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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
−12.03483285136731522522639842469, −11.15388274078272252978656911604, −9.995475655358149810052775238567, −8.624396951468068564541425058943, −7.83630286714408935051368940875, −6.36882560552128221208520100877, −5.83196721817750222205306545404, −4.79808628670086740297392275250, −3.58032899241007717443825845386, −1.67394581514406536289974816778,
1.96554194397238350990804277236, 3.39264288156590478348342924653, 4.63284482831721960809009448779, 5.60926746588969588490660937658, 6.76820706159290407196838688545, 7.29319501371539943111898220055, 9.456753879696228798638628567648, 10.07126602143981105767447275208, 10.92705710494336995064850979108, 11.89226712250151306276149785889
