L(s) = 1 | + (−1.02e3 + 592. i)5-s + (2.36e3 + 386. i)7-s + (1.80e4 + 1.03e4i)11-s + 2.00e4·13-s + (−4.81e4 − 2.77e4i)17-s + (1.04e5 + 1.81e5i)19-s + (−1.94e5 + 1.12e5i)23-s + (5.05e5 − 8.76e5i)25-s − 6.05e5i·29-s + (8.23e5 − 1.42e6i)31-s + (−2.65e6 + 1.00e6i)35-s + (1.81e5 + 3.14e5i)37-s − 5.07e6i·41-s + 1.81e6·43-s + (2.07e6 − 1.19e6i)47-s + ⋯ |
L(s) = 1 | + (−1.64 + 0.947i)5-s + (0.986 + 0.160i)7-s + (1.22 + 0.710i)11-s + 0.700·13-s + (−0.576 − 0.332i)17-s + (0.804 + 1.39i)19-s + (−0.694 + 0.400i)23-s + (1.29 − 2.24i)25-s − 0.855i·29-s + (0.891 − 1.54i)31-s + (−1.77 + 0.671i)35-s + (0.0969 + 0.167i)37-s − 1.79i·41-s + 0.530·43-s + (0.425 − 0.245i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.022518923\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.022518923\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.36e3 - 386. i)T \) |
good | 5 | \( 1 + (1.02e3 - 592. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-1.80e4 - 1.03e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.00e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (4.81e4 + 2.77e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.04e5 - 1.81e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.94e5 - 1.12e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 6.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-8.23e5 + 1.42e6i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.81e5 - 3.14e5i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 5.07e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.81e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (-2.07e6 + 1.19e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-9.71e6 - 5.60e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-2.98e5 - 1.72e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-6.29e6 - 1.09e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-3.60e6 + 6.23e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.09e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.81e7 - 3.13e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.31e5 + 1.61e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 7.69e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-5.74e7 + 3.31e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.99e7T + 7.83e15T^{2} \) |
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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
−11.06694537063243681520905724909, −9.980705491245891604380367229317, −8.632341350626661477628149018728, −7.77826477789653206772486795699, −7.11782188510507535619774592997, −5.90116193258141197184217249756, −4.16034689114251353703118803932, −3.89532287637634711601153172114, −2.29401945437622908351377869282, −0.886923933191772858750842653294,
0.64762704619561178954128088815, 1.31041049868092023900619536749, 3.34178170358955603483160319879, 4.28493776087662783381013589588, 4.98968340058334212493018028011, 6.58137677896249953231781250511, 7.67534874736078673145726622727, 8.607959550235841559750150289522, 8.940877321007232705781636401616, 10.79910708177108415094122490650