| L(s) = 1 | + 11.2i·2-s − 77.9i·3-s + 129.·4-s − 278. i·5-s + 875.·6-s − 3.16e3i·7-s + 4.33e3i·8-s + 479.·9-s + 3.12e3·10-s − 1.63e4·11-s − 1.01e4i·12-s − 529.·13-s + 3.54e4·14-s − 2.16e4·15-s − 1.53e4·16-s − 1.22e5·17-s + ⋯ |
| L(s) = 1 | + 0.701i·2-s − 0.962i·3-s + 0.507·4-s − 0.444i·5-s + 0.675·6-s − 1.31i·7-s + 1.05i·8-s + 0.0730·9-s + 0.312·10-s − 1.11·11-s − 0.488i·12-s − 0.0185·13-s + 0.923·14-s − 0.428·15-s − 0.234·16-s − 1.46·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.08199 - 1.28688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08199 - 1.28688i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.86e5 - 3.36e6i)T \) |
| good | 2 | \( 1 - 11.2iT - 256T^{2} \) |
| 3 | \( 1 + 77.9iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 278. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 3.16e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.63e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 529.T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.22e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.26e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 3.32e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.14e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 7.37e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.37e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 5.06e6T + 7.98e12T^{2} \) |
| 47 | \( 1 + 3.40e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 8.04e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.77e7T + 1.46e14T^{2} \) |
| 61 | \( 1 - 8.79e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 2.55e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.00e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 7.54e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.30e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.73e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 3.53e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.12e8T + 7.83e15T^{2} \) |
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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
−13.58839754008003014752885881654, −13.12833386931052635991839429037, −11.54595688486700018601911812657, −10.36422749828244714568014600737, −8.312729079154963208812443697602, −7.31364937395933833907658110037, −6.52722608175218914698479468641, −4.71340525251480943534356723244, −2.26635127261639861780204296217, −0.59520557299264531433888982469,
2.11551654132909601522400149813, 3.30964323049856846357641116668, 5.09588201444298230055405919432, 6.74030582042873742021249193014, 8.663589772848962489218868004827, 10.07020674262189388443675470914, 10.77297328146748040139490777818, 11.95469865499922234084605751914, 13.05345659558603419440903888000, 14.98851410617753441456478708159
