| L(s) = 1 | + (6.39 + 6.39i)2-s − 92.4i·3-s − 174. i·4-s + (596. − 596. i)5-s + (590. − 590. i)6-s − 3.52e3·7-s + (2.75e3 − 2.75e3i)8-s − 1.98e3·9-s + 7.62e3·10-s + 1.04e4i·11-s − 1.61e4·12-s + (−1.70e4 + 1.70e4i)13-s + (−2.25e4 − 2.25e4i)14-s + (−5.51e4 − 5.51e4i)15-s − 9.46e3·16-s + (−2.98e4 + 2.98e4i)17-s + ⋯ |
| L(s) = 1 | + (0.399 + 0.399i)2-s − 1.14i·3-s − 0.680i·4-s + (0.954 − 0.954i)5-s + (0.455 − 0.455i)6-s − 1.46·7-s + (0.671 − 0.671i)8-s − 0.302·9-s + 0.762·10-s + 0.716i·11-s − 0.777·12-s + (−0.597 + 0.597i)13-s + (−0.586 − 0.586i)14-s + (−1.08 − 1.08i)15-s − 0.144·16-s + (−0.357 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.642379 - 1.87702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642379 - 1.87702i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (3.87e5 - 1.83e6i)T \) |
| good | 2 | \( 1 + (-6.39 - 6.39i)T + 256iT^{2} \) |
| 3 | \( 1 + 92.4iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-596. + 596. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 + 3.52e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.04e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (1.70e4 - 1.70e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (2.98e4 - 2.98e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (-6.59e4 + 6.59e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (-3.31e5 + 3.31e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (8.09e5 + 8.09e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (-9.17e5 - 9.17e5i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 + 3.95e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (2.11e6 - 2.11e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 2.44e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 2.90e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-9.62e6 + 9.62e6i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (4.47e6 + 4.47e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 - 7.55e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 7.76e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.51e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (-3.90e7 + 3.90e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 - 2.11e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-2.26e7 - 2.26e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (1.06e8 - 1.06e8i)T - 7.83e15iT^{2} \) |
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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
−13.70843662184482938212131002457, −13.13734388708768172783325609684, −12.35350475884053472333383694488, −10.06684237547055178644944185474, −9.198145112303978077984409003394, −6.99985194976720589037438674098, −6.30013696870033832954253632161, −4.83838831932384074406367042926, −2.05392356337937676617191566312, −0.68821664065813348916112356494,
2.81951065233577001722884401571, 3.57630684333372852607604241834, 5.49108309057495065295004937119, 7.10934828139538148351054224091, 9.295084964422753217731336634666, 10.14217351974603898264157928821, 11.20967359645071390863754325034, 12.88203608625979589206354288192, 13.70898527170820152878614771345, 15.08244526584940230020388647121
