| L(s) = 1 | + (18.1 + 18.1i)3-s + (−577. − 239. i)5-s + (−190. + 190. i)7-s − 5.90e3i·9-s − 1.90e4·11-s + (−2.79e4 − 2.79e4i)13-s + (−6.11e3 − 1.48e4i)15-s + (−2.25e4 + 2.25e4i)17-s − 6.34e4i·19-s − 6.91e3·21-s + (1.96e5 + 1.96e5i)23-s + (2.75e5 + 2.76e5i)25-s + (2.25e5 − 2.25e5i)27-s + 5.95e5i·29-s + 2.49e5·31-s + ⋯ |
| L(s) = 1 | + (0.223 + 0.223i)3-s + (−0.923 − 0.383i)5-s + (−0.0794 + 0.0794i)7-s − 0.899i·9-s − 1.29·11-s + (−0.976 − 0.976i)13-s + (−0.120 − 0.292i)15-s + (−0.269 + 0.269i)17-s − 0.486i·19-s − 0.0355·21-s + (0.703 + 0.703i)23-s + (0.706 + 0.708i)25-s + (0.425 − 0.425i)27-s + 0.842i·29-s + 0.269·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.160499 - 0.497780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160499 - 0.497780i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (577. + 239. i)T \) |
| good | 3 | \( 1 + (-18.1 - 18.1i)T + 6.56e3iT^{2} \) |
| 7 | \( 1 + (190. - 190. i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + 1.90e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (2.79e4 + 2.79e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (2.25e4 - 2.25e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + 6.34e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.96e5 - 1.96e5i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 - 5.95e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 2.49e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.29e5 + 2.29e5i)T - 3.51e12iT^{2} \) |
| 41 | \( 1 + 4.30e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (3.59e6 + 3.59e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + (-6.19e6 + 6.19e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 + (2.53e5 + 2.53e5i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 - 7.27e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 3.24e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + (-2.68e7 + 2.68e7i)T - 4.06e14iT^{2} \) |
| 71 | \( 1 + 3.70e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (2.76e7 + 2.76e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 2.03e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (2.23e7 + 2.23e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 - 2.64e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-9.78e7 + 9.78e7i)T - 7.83e15iT^{2} \) |
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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
−15.58118444080522448601792019952, −15.09699391259217704577949392727, −13.10333693655970366376885900173, −12.01849616282150292176532531496, −10.39056039671122018605997034491, −8.770135654491642571136273554352, −7.37450631697838166744886584192, −5.06645271533106482217520107901, −3.18953739523807708609057166360, −0.24790021959691861086426109611,
2.58060495656363156910575741379, 4.72593194009678371211659790338, 7.12010417819623631121098485997, 8.227293355123380201394781639920, 10.26278201855417504318919215660, 11.58862006042399688622710545264, 13.04701696319922408628872808201, 14.40192216612212106740507547190, 15.68892533557704109311336471141, 16.78696976389403966715940362807
