| L(s) = 1 | − 23.5i·2-s + 46.0i·3-s − 297.·4-s − 373. i·5-s + 1.08e3·6-s − 2.48e3i·7-s + 983. i·8-s + 4.43e3·9-s − 8.79e3·10-s + 7.40e3·11-s − 1.37e4i·12-s − 3.62e4·13-s − 5.85e4·14-s + 1.72e4·15-s − 5.30e4·16-s − 8.88e4·17-s + ⋯ |
| L(s) = 1 | − 1.47i·2-s + 0.568i·3-s − 1.16·4-s − 0.598i·5-s + 0.836·6-s − 1.03i·7-s + 0.240i·8-s + 0.676·9-s − 0.879·10-s + 0.505·11-s − 0.661i·12-s − 1.27·13-s − 1.52·14-s + 0.340·15-s − 0.810·16-s − 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.379940 + 1.03273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379940 + 1.03273i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.60e6 + 2.21e6i)T \) |
| good | 2 | \( 1 + 23.5iT - 256T^{2} \) |
| 3 | \( 1 - 46.0iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 373. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.48e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 7.40e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + 3.62e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.88e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 4.18e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 2.94e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 4.31e4iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 5.57e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.70e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.14e6T + 7.98e12T^{2} \) |
| 47 | \( 1 + 7.21e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + 8.22e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 5.76e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 1.24e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 2.60e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.21e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.56e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 2.11e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 5.69e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 6.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.82e7T + 7.83e15T^{2} \) |
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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
−13.07522236012820334962799138191, −12.25406727748817378873785835506, −10.92523734758186824055592723195, −10.05182263132382033705273548638, −9.148776513030406393875024601663, −7.15892854099430990370142234741, −4.64148954479795631686176841437, −3.83059422191901187196442584606, −1.88422791563667698349543030388, −0.40260477446746573872087862472,
2.27363017273292034608518438950, 4.80069829948449054164389415072, 6.34346503936917476332078329627, 7.07105734798809488902304062195, 8.313082514109970476274144726456, 9.657655978883245742518976634365, 11.57653046408702644584348665951, 12.82436363703067348383073274999, 14.16529575921926630719652002680, 15.06263025475361286517487267463
