| L(s) = 1 | + (−76.4 − 26.8i)3-s + (266. + 565. i)5-s − 424. i·7-s + (5.11e3 + 4.10e3i)9-s + 1.11e4i·11-s + 2.95e4i·13-s + (−5.13e3 − 5.03e4i)15-s + 1.51e5·17-s + 3.12e4·19-s + (−1.14e4 + 3.24e4i)21-s + 1.31e5·23-s + (−2.48e5 + 3.01e5i)25-s + (−2.80e5 − 4.51e5i)27-s + 7.99e5i·29-s − 3.83e4·31-s + ⋯ |
| L(s) = 1 | + (−0.943 − 0.331i)3-s + (0.425 + 0.904i)5-s − 0.176i·7-s + (0.779 + 0.626i)9-s + 0.761i·11-s + 1.03i·13-s + (−0.101 − 0.994i)15-s + 1.80·17-s + 0.239·19-s + (−0.0586 + 0.166i)21-s + 0.468·23-s + (−0.637 + 0.770i)25-s + (−0.527 − 0.849i)27-s + 1.12i·29-s − 0.0415·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.101 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.726690839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.726690839\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (76.4 + 26.8i)T \) |
| 5 | \( 1 + (-266. - 565. i)T \) |
| good | 7 | \( 1 + 424. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.11e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.95e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.51e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 3.12e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.31e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 7.99e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.83e4T + 8.52e11T^{2} \) |
| 37 | \( 1 + 3.96e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 3.00e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 7.36e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 8.11e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 2.70e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.23e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.18e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.77e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.36e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.00e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 2.59e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 2.06e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.09e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 3.70e7iT - 7.83e15T^{2} \) |
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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
−10.86327659232888529452343395902, −10.21739005143507595475625867798, −9.278005908852934821741725510526, −7.50539154535213097822787081639, −7.03179764778039060533005123401, −5.96865091694174999613918639725, −5.02792074774743917402754636329, −3.64832454751035466469248788822, −2.14843847404692192658330709772, −1.09542588494097839367277317384,
0.53750857479985139322398094678, 1.21118337588677693399242457656, 3.08247321951919848454584522989, 4.41692745627284930673024874201, 5.66824693013617716203092402900, 5.78932613356048143978359648922, 7.49964920824186977628166519594, 8.554683837239223085248593386421, 9.699191358510998080643449223348, 10.31874474405683568943653250895
