L(s) = 1 | + 5-s − 7-s − 3·9-s + 6·11-s − 2·13-s − 3·17-s + 6·19-s − 23-s + 25-s + 3·29-s + 3·31-s − 35-s + 37-s + 9·41-s + 8·43-s − 3·45-s − 4·47-s − 6·49-s + 53-s + 6·55-s − 59-s + 8·61-s + 3·63-s − 2·65-s + 7·67-s + 5·71-s − 6·73-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 1.80·11-s − 0.554·13-s − 0.727·17-s + 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.557·29-s + 0.538·31-s − 0.169·35-s + 0.164·37-s + 1.40·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s − 6/7·49-s + 0.137·53-s + 0.809·55-s − 0.130·59-s + 1.02·61-s + 0.377·63-s − 0.248·65-s + 0.855·67-s + 0.593·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.826020854\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.826020854\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−9.370845902172061297584224195948, −8.648963889549550997042854812696, −7.66422277773345536118064776340, −6.67290253533681347908702724491, −6.18709233320869651885027817519, −5.28023148829339091268675270237, −4.26366539891827765942254240707, −3.27734500100801096074623940175, −2.32566640319124508162435295458, −0.943553537319372488144220598724,
0.943553537319372488144220598724, 2.32566640319124508162435295458, 3.27734500100801096074623940175, 4.26366539891827765942254240707, 5.28023148829339091268675270237, 6.18709233320869651885027817519, 6.67290253533681347908702724491, 7.66422277773345536118064776340, 8.648963889549550997042854812696, 9.370845902172061297584224195948