L(s) = 1 | + 24.2i·2-s + 13.8i·3-s − 333.·4-s − 427. i·5-s − 335.·6-s − 2.56e3i·7-s − 1.89e3i·8-s + 6.37e3·9-s + 1.03e4·10-s + 3.29e3·11-s − 4.61e3i·12-s + 1.78e3·13-s + 6.22e4·14-s + 5.90e3·15-s − 3.94e4·16-s + 1.61e5·17-s + ⋯ |
L(s) = 1 | + 1.51i·2-s + 0.170i·3-s − 1.30·4-s − 0.684i·5-s − 0.258·6-s − 1.06i·7-s − 0.462i·8-s + 0.970·9-s + 1.03·10-s + 0.225·11-s − 0.222i·12-s + 0.0624·13-s + 1.62·14-s + 0.116·15-s − 0.602·16-s + 1.93·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.69070 + 0.942438i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69070 + 0.942438i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-1.79e6 + 2.90e6i)T \) |
good | 2 | \( 1 - 24.2iT - 256T^{2} \) |
| 3 | \( 1 - 13.8iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 427. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.56e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 3.29e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 1.78e3T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.61e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.88e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.30e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.23e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 8.19e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.89e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 1.24e6T + 7.98e12T^{2} \) |
| 47 | \( 1 - 3.12e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.47e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.77e5T + 1.46e14T^{2} \) |
| 61 | \( 1 + 1.30e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.94e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.97e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.15e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 8.29e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.28e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 2.70e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.00e8T + 7.83e15T^{2} \) |
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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
−14.51274721036125862051821869266, −13.64655828446114191675650024637, −12.38884845289283378322800872506, −10.46205592225736311787421315362, −9.151658496384113886271785713895, −7.76937889854151367362420861387, −6.90210346680321964372905150360, −5.30043444646400366824245508457, −4.11876159361576966072406739611, −0.906506023504202229375683455935,
1.30201666516492542882992296176, 2.61167721560011482251233663844, 3.96870414845807537169645004866, 6.04344095820293799568549495437, 7.929187995691876013428163884333, 9.734338912506677957629054323442, 10.31314802488710758018209530143, 11.99793160795639495492102048110, 12.20721162037831967359768656648, 13.71526302722188978516330805189