L(s) = 1 | − 3.74·3-s + (4.58 + 2i)5-s − 2.44·7-s + 5·9-s + 14.6i·11-s − 8i·13-s + (−17.1 − 7.48i)15-s − 18.3i·17-s − 24.4i·19-s + 9.16·21-s − 26.9·23-s + (17 + 18.3i)25-s + 14.9·27-s + 18.3·29-s + 44.8i·31-s + ⋯ |
L(s) = 1 | − 1.24·3-s + (0.916 + 0.400i)5-s − 0.349·7-s + 0.555·9-s + 1.33i·11-s − 0.615i·13-s + (−1.14 − 0.498i)15-s − 1.07i·17-s − 1.28i·19-s + 0.436·21-s − 1.17·23-s + (0.680 + 0.733i)25-s + 0.554·27-s + 0.632·29-s + 1.44i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.128010035\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128010035\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-4.58 - 2i)T \) |
good | 3 | \( 1 + 3.74T + 9T^{2} \) |
| 7 | \( 1 + 2.44T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 + 24.4iT - 361T^{2} \) |
| 23 | \( 1 + 26.9T + 529T^{2} \) |
| 29 | \( 1 - 18.3T + 841T^{2} \) |
| 31 | \( 1 - 44.8iT - 961T^{2} \) |
| 37 | \( 1 + 44iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 20T + 1.68e3T^{2} \) |
| 43 | \( 1 + 11.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 32iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 64.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 78.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 91.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 44.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 11.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 10T + 7.92e3T^{2} \) |
| 97 | \( 1 + 54.9iT - 9.40e3T^{2} \) |
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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
−9.752247496736331711431636152809, −8.821908645087109676926955449656, −7.43324326274854435399926343221, −6.78154837976841649414383648901, −6.14853049723490738076071724555, −5.17690721189789095483253282394, −4.72178135874326386545340844019, −3.09406056671275354557384258785, −2.04661300867348307945596034317, −0.53926888190247004265951867034,
0.810620798183469327225169742200, 1.98919095080260174334733969318, 3.50685828506820739668986201698, 4.58170958367771962454370639274, 5.68377333545213146840088848063, 6.06608404080022419456012387660, 6.54573250857596969528261633060, 8.105469901540101087500641584484, 8.639026172054304935595684575641, 9.939797226214694281928775050410