| L(s) = 1 | + 42.6·3-s + (560. + 276. i)5-s − 869.·7-s − 4.74e3·9-s + 2.47e4i·11-s − 4.11e4i·13-s + (2.39e4 + 1.18e4i)15-s − 4.09e4i·17-s + 7.95e4i·19-s − 3.71e4·21-s + 4.28e5·23-s + (2.37e5 + 3.10e5i)25-s − 4.82e5·27-s − 1.84e5·29-s − 1.54e5i·31-s + ⋯ |
| L(s) = 1 | + 0.526·3-s + (0.896 + 0.443i)5-s − 0.362·7-s − 0.722·9-s + 1.68i·11-s − 1.44i·13-s + (0.472 + 0.233i)15-s − 0.489i·17-s + 0.610i·19-s − 0.190·21-s + 1.53·23-s + (0.607 + 0.794i)25-s − 0.907·27-s − 0.261·29-s − 0.167i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.287669824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287669824\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-560. - 276. i)T \) |
| good | 3 | \( 1 - 42.6T + 6.56e3T^{2} \) |
| 7 | \( 1 + 869.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.47e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.11e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.09e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 7.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 4.28e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.84e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.54e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.72e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.00e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 1.43e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.98e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.97e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.65e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 6.73e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 4.99e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 2.07e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.31e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 7.49e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 8.75e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.14e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 8.74e7iT - 7.83e15T^{2} \) |
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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
−10.34272727108607765392026646523, −9.786242680870094868577111633709, −8.917828373062360789145191213568, −7.75528384277697045117536963707, −6.89421877550759590625798936585, −5.77469262854184620463034181360, −4.87249578291037199703473236808, −3.21915641331330956045616481694, −2.60581080002698636644084264061, −1.41741445164332752367858300337,
0.21709518189261928996485261273, 1.47404582273366791056348482181, 2.64628439751858567152607653201, 3.56124301365489699015279759804, 5.00946138481640910715006643469, 5.98799060345422126751075524581, 6.78015311764701071472576627676, 8.330650526375587379231771459826, 8.950691726408931392411379613364, 9.496104371930505225053162512904
