| L(s) = 1 | − 4·2-s + 16·4-s + 172·7-s − 64·8-s − 132·11-s + 946·13-s − 688·14-s + 256·16-s − 222·17-s + 500·19-s + 528·22-s + 3.56e3·23-s − 3.78e3·26-s + 2.75e3·28-s − 2.19e3·29-s + 2.31e3·31-s − 1.02e3·32-s + 888·34-s + 1.12e4·37-s − 2.00e3·38-s − 1.24e3·41-s − 2.06e4·43-s − 2.11e3·44-s − 1.42e4·46-s + 6.58e3·47-s + 1.27e4·49-s + 1.51e4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.32·7-s − 0.353·8-s − 0.328·11-s + 1.55·13-s − 0.938·14-s + 1/4·16-s − 0.186·17-s + 0.317·19-s + 0.232·22-s + 1.40·23-s − 1.09·26-s + 0.663·28-s − 0.483·29-s + 0.432·31-s − 0.176·32-s + 0.131·34-s + 1.35·37-s − 0.224·38-s − 0.115·41-s − 1.70·43-s − 0.164·44-s − 0.993·46-s + 0.435·47-s + 0.760·49-s + 0.776·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.052020377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052020377\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 172 T + p^{5} T^{2} \) |
| 11 | \( 1 + 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 946 T + p^{5} T^{2} \) |
| 17 | \( 1 + 222 T + p^{5} T^{2} \) |
| 19 | \( 1 - 500 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 + 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 + 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 + 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 + 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 + 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 + 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 - 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 - 78382 T + p^{5} T^{2} \) |
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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
−10.39250654964741321462581433129, −9.224827628100217360775370840755, −8.428379968879557506191922502682, −7.81589293538110726129808019001, −6.72536580130094358593486761015, −5.61819415783548099646915377980, −4.56305015137502829056681526315, −3.17779705647136816637920844402, −1.76776098091848921514237319949, −0.883626221114323510457457799825,
0.883626221114323510457457799825, 1.76776098091848921514237319949, 3.17779705647136816637920844402, 4.56305015137502829056681526315, 5.61819415783548099646915377980, 6.72536580130094358593486761015, 7.81589293538110726129808019001, 8.428379968879557506191922502682, 9.224827628100217360775370840755, 10.39250654964741321462581433129
