| L(s) = 1 | + (−2 + 2i)5-s − 4.24i·7-s + (−2.82 + 2.82i)11-s + (3 + 3i)13-s + 6·17-s + (−1.41 − 1.41i)19-s − 2.82i·23-s − 3i·25-s + (−4 − 4i)29-s − 4.24·31-s + (8.48 + 8.48i)35-s + (3 − 3i)37-s + 10i·41-s + (−4.24 + 4.24i)43-s − 2.82·47-s + ⋯ |
| L(s) = 1 | + (−0.894 + 0.894i)5-s − 1.60i·7-s + (−0.852 + 0.852i)11-s + (0.832 + 0.832i)13-s + 1.45·17-s + (−0.324 − 0.324i)19-s − 0.589i·23-s − 0.600i·25-s + (−0.742 − 0.742i)29-s − 0.762·31-s + (1.43 + 1.43i)35-s + (0.493 − 0.493i)37-s + 1.56i·41-s + (−0.646 + 0.646i)43-s − 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8410156556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8410156556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2 - 2i)T - 5iT^{2} \) |
| 7 | \( 1 + 4.24iT - 7T^{2} \) |
| 11 | \( 1 + (2.82 - 2.82i)T - 11iT^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (1.41 + 1.41i)T + 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (4 + 4i)T + 29iT^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + (-3 + 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + (4.24 - 4.24i)T - 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 - 59iT^{2} \) |
| 61 | \( 1 + (-3 - 3i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.82 - 2.82i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−9.460782334612069539042251376314, −7.963375893695145091242750927877, −7.83047922738050491503937253599, −6.98269244286854033136201617732, −6.46787669523686087911171125593, −5.16784833704585551655689096009, −4.10765781821088163764500017615, −3.76489249449833151399478317322, −2.66509754649516254128699670644, −1.20732504073449967939306802557,
0.31732378595242006805976043831, 1.73884290339946242779963227575, 3.14274292787743549056078545070, 3.62421376643498998759795845427, 5.11198517483063446117853502887, 5.46575774805932600455139064870, 6.12037577768347044742358463764, 7.59648121307244308576720067545, 8.120962446745646834497444679672, 8.641717433604495954971281477584
