| L(s) = 1 | − 4·2-s + 16·4-s + 66·5-s + 176·7-s − 64·8-s − 264·10-s + 60·11-s − 658·13-s − 704·14-s + 256·16-s + 414·17-s + 956·19-s + 1.05e3·20-s − 240·22-s − 600·23-s + 1.23e3·25-s + 2.63e3·26-s + 2.81e3·28-s − 5.57e3·29-s − 3.59e3·31-s − 1.02e3·32-s − 1.65e3·34-s + 1.16e4·35-s − 8.45e3·37-s − 3.82e3·38-s − 4.22e3·40-s − 1.91e4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.18·5-s + 1.35·7-s − 0.353·8-s − 0.834·10-s + 0.149·11-s − 1.07·13-s − 0.959·14-s + 1/4·16-s + 0.347·17-s + 0.607·19-s + 0.590·20-s − 0.105·22-s − 0.236·23-s + 0.393·25-s + 0.763·26-s + 0.678·28-s − 1.23·29-s − 0.671·31-s − 0.176·32-s − 0.245·34-s + 1.60·35-s − 1.01·37-s − 0.429·38-s − 0.417·40-s − 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.253113366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.253113366\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 66 T + p^{5} T^{2} \) |
| 7 | \( 1 - 176 T + p^{5} T^{2} \) |
| 11 | \( 1 - 60 T + p^{5} T^{2} \) |
| 13 | \( 1 + 658 T + p^{5} T^{2} \) |
| 17 | \( 1 - 414 T + p^{5} T^{2} \) |
| 19 | \( 1 - 956 T + p^{5} T^{2} \) |
| 23 | \( 1 + 600 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 + 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 - 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 - 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 + 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 + 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 - 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 - 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 - 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 - 166082 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.60143514747107855641847853621, −16.87164772888336234276489010211, −14.96418520544948281580193268289, −13.86991612116280703195375597256, −11.96966521577168841456746050653, −10.46840057583256842149293444215, −9.176764611080178693852768714264, −7.50027778823224844564042209893, −5.41916015996980546828401580900, −1.85679263470097699384725873814,
1.85679263470097699384725873814, 5.41916015996980546828401580900, 7.50027778823224844564042209893, 9.176764611080178693852768714264, 10.46840057583256842149293444215, 11.96966521577168841456746050653, 13.86991612116280703195375597256, 14.96418520544948281580193268289, 16.87164772888336234276489010211, 17.60143514747107855641847853621
