| L(s) = 1 | + (−2 + 2i)2-s + (0.493 − 1.19i)3-s − 8i·4-s + (−21.5 − 21.5i)5-s + (1.39 + 3.36i)6-s + (−12.4 + 30.0i)7-s + (16 + 16i)8-s + (56.1 + 56.1i)9-s + 86.2·10-s + (100. + 41.7i)11-s + (−9.52 − 3.94i)12-s + (−11.1 + 26.8i)13-s + (−35.1 − 84.9i)14-s + (−36.3 + 15.0i)15-s − 64·16-s + (60.8 + 146. i)17-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.5i)2-s + (0.0547 − 0.132i)3-s − 0.5i·4-s + (−0.862 − 0.862i)5-s + (0.0387 + 0.0935i)6-s + (−0.253 + 0.612i)7-s + (0.250 + 0.250i)8-s + (0.692 + 0.692i)9-s + 0.862·10-s + (0.832 + 0.344i)11-s + (−0.0661 − 0.0273i)12-s + (−0.0658 + 0.158i)13-s + (−0.179 − 0.433i)14-s + (−0.161 + 0.0668i)15-s − 0.250·16-s + (0.210 + 0.507i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.765128 + 0.673234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765128 + 0.673234i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 2i)T \) |
| 41 | \( 1 + (-1.24e3 + 1.12e3i)T \) |
| good | 3 | \( 1 + (-0.493 + 1.19i)T + (-57.2 - 57.2i)T^{2} \) |
| 5 | \( 1 + (21.5 + 21.5i)T + 625iT^{2} \) |
| 7 | \( 1 + (12.4 - 30.0i)T + (-1.69e3 - 1.69e3i)T^{2} \) |
| 11 | \( 1 + (-100. - 41.7i)T + (1.03e4 + 1.03e4i)T^{2} \) |
| 13 | \( 1 + (11.1 - 26.8i)T + (-2.01e4 - 2.01e4i)T^{2} \) |
| 17 | \( 1 + (-60.8 - 146. i)T + (-5.90e4 + 5.90e4i)T^{2} \) |
| 19 | \( 1 + (-186. - 450. i)T + (-9.21e4 + 9.21e4i)T^{2} \) |
| 23 | \( 1 - 362. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + (0.202 - 0.489i)T + (-5.00e5 - 5.00e5i)T^{2} \) |
| 31 | \( 1 - 838. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.21e3T + 1.87e6T^{2} \) |
| 43 | \( 1 + (-239. + 239. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (434. + 1.04e3i)T + (-3.45e6 + 3.45e6i)T^{2} \) |
| 53 | \( 1 + (-1.94e3 - 807. i)T + (5.57e6 + 5.57e6i)T^{2} \) |
| 59 | \( 1 - 4.07e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + (2.77e3 - 2.77e3i)T - 1.38e7iT^{2} \) |
| 67 | \( 1 + (1.65e3 + 3.98e3i)T + (-1.42e7 + 1.42e7i)T^{2} \) |
| 71 | \( 1 + (1.13e3 - 2.74e3i)T + (-1.79e7 - 1.79e7i)T^{2} \) |
| 73 | \( 1 + (3.86e3 - 3.86e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + (7.81e3 + 3.23e3i)T + (2.75e7 + 2.75e7i)T^{2} \) |
| 83 | \( 1 - 1.04e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (4.72e3 - 1.14e4i)T + (-4.43e7 - 4.43e7i)T^{2} \) |
| 97 | \( 1 + (1.09e4 - 4.52e3i)T + (6.25e7 - 6.25e7i)T^{2} \) |
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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
−13.95871937003784435556614656468, −12.55421564684062810451588455603, −11.89590376696230258791570291151, −10.31284318828740018382790298231, −9.127170441587816837391342446933, −8.154054714968859612941384094862, −7.12916107789683811130449418619, −5.52623381162032317032224387918, −4.08904803074101865427243484795, −1.46036369073184139852565591936,
0.67104135740399952539106568895, 3.12466477276935279945829337609, 4.18212407541495438469342088408, 6.71176216899040302906690686452, 7.47791831484372836179374849287, 9.031339363830901065959796526011, 10.07946383275380962592929161455, 11.16351927747577927497357615443, 11.93244484827965657364159607439, 13.21980136366129180748217660418
