L(s) = 1 | + (0.379 − 5.64i)2-s + (−20.6 + 20.6i)3-s + (−31.7 − 4.28i)4-s + (−28.3 − 28.3i)5-s + (108. + 124. i)6-s + 55.5i·7-s + (−36.2 + 177. i)8-s − 609. i·9-s + (−170. + 149. i)10-s + (−137. − 137. i)11-s + (743. − 566. i)12-s + (−574. + 574. i)13-s + (313. + 21.0i)14-s + 1.16e3·15-s + (987. + 271. i)16-s + 320.·17-s + ⋯ |
L(s) = 1 | + (0.0671 − 0.997i)2-s + (−1.32 + 1.32i)3-s + (−0.990 − 0.133i)4-s + (−0.506 − 0.506i)5-s + (1.23 + 1.41i)6-s + 0.428i·7-s + (−0.200 + 0.979i)8-s − 2.50i·9-s + (−0.539 + 0.471i)10-s + (−0.342 − 0.342i)11-s + (1.48 − 1.13i)12-s + (−0.942 + 0.942i)13-s + (0.427 + 0.0287i)14-s + 1.34·15-s + (0.964 + 0.265i)16-s + 0.269·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0392581 + 0.114355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0392581 + 0.114355i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.379 + 5.64i)T \) |
good | 3 | \( 1 + (20.6 - 20.6i)T - 243iT^{2} \) |
| 5 | \( 1 + (28.3 + 28.3i)T + 3.12e3iT^{2} \) |
| 7 | \( 1 - 55.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (137. + 137. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (574. - 574. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 - 320.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (858. - 858. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 825. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-333. + 333. i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 + 8.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (3.87e3 + 3.87e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 6.54e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (5.62e3 + 5.62e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 3.12e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.81e4 - 1.81e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (5.53e3 + 5.53e3i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (701. - 701. i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (1.44e4 - 1.44e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.36e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.77e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.89e4 + 7.89e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 2.59e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 3.91e4T + 8.58e9T^{2} \) |
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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
−18.59469512825189262584961550419, −17.16058893212688426841699678820, −16.21016417500276715246403851667, −14.74021172025697329372418588033, −12.39375020720885114303269852902, −11.59378641697458087288877766885, −10.34401874728858061143161799542, −9.074455794573723310149484051954, −5.42989290303281382266806998268, −4.13107145259953262085178448441,
0.11184407268388551157553779816, 5.27173199795994209428123613059, 6.90920566287481238749582251026, 7.70811720523750799451439408686, 10.62896991624589694633054934852, 12.30431069499245901605012380777, 13.29642534992990512947619069901, 14.98241725836097394494467109006, 16.58473769457496626968517778687, 17.51903179981328052097510362912