L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 2·11-s − 6·13-s − 14-s + 16-s − 5·17-s + 6·19-s + 2·22-s + 3·23-s + 6·26-s + 28-s − 6·29-s − 4·31-s − 32-s + 5·34-s + 4·37-s − 6·38-s − 41-s − 7·43-s − 2·44-s − 3·46-s − 2·47-s + 49-s − 6·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.603·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.21·17-s + 1.37·19-s + 0.426·22-s + 0.625·23-s + 1.17·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.857·34-s + 0.657·37-s − 0.973·38-s − 0.156·41-s − 1.06·43-s − 0.301·44-s − 0.442·46-s − 0.291·47-s + 1/7·49-s − 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9048098416\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9048098416\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.51424701213402463478009259666, −7.31439834956837859890623634134, −6.59923219478333336144782816159, −5.46536684048708675333442394408, −5.14611054327382513682929294415, −4.27628350833263709224070385933, −3.20068535088647466705891775893, −2.45015843339228017740159148031, −1.76124442168877179572682608602, −0.48919302319906811019610924684,
0.48919302319906811019610924684, 1.76124442168877179572682608602, 2.45015843339228017740159148031, 3.20068535088647466705891775893, 4.27628350833263709224070385933, 5.14611054327382513682929294415, 5.46536684048708675333442394408, 6.59923219478333336144782816159, 7.31439834956837859890623634134, 7.51424701213402463478009259666