| L(s) = 1 | − 4·2-s + 16·4-s − 33·5-s + 59·7-s − 64·8-s + 132·10-s − 147·11-s + 836·13-s − 236·14-s + 256·16-s + 1.08e3·17-s + 2.88e3·19-s − 528·20-s + 588·22-s + 4.38e3·23-s − 2.03e3·25-s − 3.34e3·26-s + 944·28-s − 1.86e3·29-s − 3.29e3·31-s − 1.02e3·32-s − 4.32e3·34-s − 1.94e3·35-s − 3.95e3·37-s − 1.15e4·38-s + 2.11e3·40-s + 2.05e4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.590·5-s + 0.455·7-s − 0.353·8-s + 0.417·10-s − 0.366·11-s + 1.37·13-s − 0.321·14-s + 1/4·16-s + 0.906·17-s + 1.83·19-s − 0.295·20-s + 0.259·22-s + 1.72·23-s − 0.651·25-s − 0.970·26-s + 0.227·28-s − 0.412·29-s − 0.615·31-s − 0.176·32-s − 0.640·34-s − 0.268·35-s − 0.475·37-s − 1.29·38-s + 0.208·40-s + 1.91·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.211547983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211547983\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 33 T + p^{5} T^{2} \) |
| 7 | \( 1 - 59 T + p^{5} T^{2} \) |
| 11 | \( 1 + 147 T + p^{5} T^{2} \) |
| 13 | \( 1 - 836 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1080 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2882 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4386 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1866 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3295 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3958 T + p^{5} T^{2} \) |
| 41 | \( 1 - 20586 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8770 T + p^{5} T^{2} \) |
| 47 | \( 1 + 12666 T + p^{5} T^{2} \) |
| 53 | \( 1 - 9621 T + p^{5} T^{2} \) |
| 59 | \( 1 - 21468 T + p^{5} T^{2} \) |
| 61 | \( 1 - 36248 T + p^{5} T^{2} \) |
| 67 | \( 1 - 5174 T + p^{5} T^{2} \) |
| 71 | \( 1 + 63720 T + p^{5} T^{2} \) |
| 73 | \( 1 - 57953 T + p^{5} T^{2} \) |
| 79 | \( 1 - 16448 T + p^{5} T^{2} \) |
| 83 | \( 1 + 69267 T + p^{5} T^{2} \) |
| 89 | \( 1 - 54198 T + p^{5} T^{2} \) |
| 97 | \( 1 + 132961 T + p^{5} T^{2} \) |
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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
−14.53013194832090314433279655599, −13.15918228117496498552336493405, −11.68877047521118145735587908746, −10.95618719034647806460054476205, −9.513972257881540988169841495493, −8.245626265256390208611842983929, −7.27523430156205761591672829099, −5.47510344416824489863302581580, −3.39736047609614033940733874289, −1.10221327952401270641430760415,
1.10221327952401270641430760415, 3.39736047609614033940733874289, 5.47510344416824489863302581580, 7.27523430156205761591672829099, 8.245626265256390208611842983929, 9.513972257881540988169841495493, 10.95618719034647806460054476205, 11.68877047521118145735587908746, 13.15918228117496498552336493405, 14.53013194832090314433279655599
