| L(s) = 1 | + 4i·2-s + (15.3 − 15.3i)3-s − 16·4-s + 59.4i·5-s + (61.2 + 61.2i)6-s + (−111. + 111. i)7-s − 64i·8-s − 226. i·9-s − 237.·10-s + (13.6 − 13.6i)11-s + (−245. + 245. i)12-s + (−546. + 546. i)13-s + (−446. − 446. i)14-s + (909. + 909. i)15-s + 256·16-s + (475. + 475. i)17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.982 − 0.982i)3-s − 0.5·4-s + 1.06i·5-s + (0.694 + 0.694i)6-s + (−0.861 + 0.861i)7-s − 0.353i·8-s − 0.930i·9-s − 0.751·10-s + (0.0340 − 0.0340i)11-s + (−0.491 + 0.491i)12-s + (−0.896 + 0.896i)13-s + (−0.609 − 0.609i)14-s + (1.04 + 1.04i)15-s + 0.250·16-s + (0.398 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.770048 + 1.45656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.770048 + 1.45656i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 41 | \( 1 + (8.66e3 + 6.38e3i)T \) |
| good | 3 | \( 1 + (-15.3 + 15.3i)T - 243iT^{2} \) |
| 5 | \( 1 - 59.4iT - 3.12e3T^{2} \) |
| 7 | \( 1 + (111. - 111. i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + (-13.6 + 13.6i)T - 1.61e5iT^{2} \) |
| 13 | \( 1 + (546. - 546. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-475. - 475. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + (-1.57e3 - 1.57e3i)T + 2.47e6iT^{2} \) |
| 23 | \( 1 + 2.91e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-101. + 101. i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 + 5.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.26e3T + 6.93e7T^{2} \) |
| 43 | \( 1 + 7.33e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-2.00e4 - 2.00e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.61e4 + 1.61e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 648.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.39e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.52e4 - 2.52e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + (3.64e4 - 3.64e4i)T - 1.80e9iT^{2} \) |
| 73 | \( 1 + 3.87e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + (1.31e4 - 1.31e4i)T - 3.07e9iT^{2} \) |
| 83 | \( 1 + 1.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-7.41e4 + 7.41e4i)T - 5.58e9iT^{2} \) |
| 97 | \( 1 + (-2.52e4 - 2.52e4i)T + 8.58e9iT^{2} \) |
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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
−14.21229321864059533489819468516, −12.82411386361285162231917005539, −11.96487222942284666799800698987, −10.03865170777829630210850872833, −9.024694500235484464160174579623, −7.74534210421848425030681601626, −6.95059330504879462464363240637, −5.87295698860591482757634781551, −3.43640545088062157879841004662, −2.16800714236755013356642692667,
0.60699639519476208479507409855, 2.87737043909636677348888150370, 4.00148703954988224645404524477, 5.16152838939050386147972808102, 7.56223986376989930623514743480, 8.889653000625923815732933258401, 9.697930957328725161954411390058, 10.33491095143105373353833333507, 12.00434197523062107205545523833, 13.08279780244023634598213838921
