| L(s) = 1 | + (9.79 − 5.65i)2-s + (63.9 − 110. i)4-s + (115. + 66.4i)5-s + (−1.06e3 − 1.83e3i)7-s − 1.44e3i·8-s + 1.50e3·10-s + (−1.68e4 + 9.71e3i)11-s + (1.95e4 − 3.38e4i)13-s + (−2.07e4 − 1.19e4i)14-s + (−8.19e3 − 1.41e4i)16-s − 3.10e4i·17-s − 1.03e5·19-s + (1.47e4 − 8.50e3i)20-s + (−1.09e5 + 1.90e5i)22-s + (−3.01e5 − 1.74e5i)23-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.184 + 0.106i)5-s + (−0.441 − 0.764i)7-s − 0.353i·8-s + 0.150·10-s + (−1.14 + 0.663i)11-s + (0.683 − 1.18i)13-s + (−0.540 − 0.312i)14-s + (−0.125 − 0.216i)16-s − 0.371i·17-s − 0.791·19-s + (0.0920 − 0.0531i)20-s + (−0.469 + 0.812i)22-s + (−1.07 − 0.622i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.345265 - 1.47666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.345265 - 1.47666i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.79 + 5.65i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-115. - 66.4i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.06e3 + 1.83e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.68e4 - 9.71e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.95e4 + 3.38e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 3.10e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.03e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (3.01e5 + 1.74e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.55e5 + 2.05e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (4.93e5 - 8.54e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.35e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (3.11e6 + 1.79e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.12e6 - 1.95e6i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-5.73e6 + 3.31e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + 6.44e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.83e6 - 2.79e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-9.61e6 - 1.66e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.63e7 + 2.83e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.42e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.34e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.83e7 - 3.17e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-4.67e7 + 2.69e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 6.24e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-4.05e7 - 7.02e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−13.15486489739808577527309127851, −12.26754485089681097027139614020, −10.55638400648700229210208755483, −10.17905486064479848376424132106, −8.183971718445896966772057936256, −6.74334710692334169006980719044, −5.33973248044325078575624014797, −3.84358341136745201279735408063, −2.36260086224677433254910446556, −0.39849080465710158402385456623,
2.15004733638707107422828086927, 3.75011634519406586205071752820, 5.44439097457787329328680727063, 6.39690192005662544518246343140, 8.045454543058844909383938112271, 9.220742253165072232574282784612, 10.82957433474364858685857750075, 12.03859837868808500406718398310, 13.16185022482512499290066303046, 13.95805682368218767748765545104
