| L(s) = 1 | − 1.56·3-s + 5-s − 1.56·7-s − 0.561·9-s − 11-s + 2·13-s − 1.56·15-s + 3.56·17-s + 1.56·19-s + 2.43·21-s + 3.12·23-s + 25-s + 5.56·27-s − 2.68·29-s − 2.43·31-s + 1.56·33-s − 1.56·35-s + 6.68·37-s − 3.12·39-s + 2·41-s + 6.24·43-s − 0.561·45-s + 4.87·47-s − 4.56·49-s − 5.56·51-s + 0.438·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.447·5-s − 0.590·7-s − 0.187·9-s − 0.301·11-s + 0.554·13-s − 0.403·15-s + 0.863·17-s + 0.358·19-s + 0.532·21-s + 0.651·23-s + 0.200·25-s + 1.07·27-s − 0.498·29-s − 0.437·31-s + 0.271·33-s − 0.263·35-s + 1.09·37-s − 0.500·39-s + 0.312·41-s + 0.952·43-s − 0.0837·45-s + 0.711·47-s − 0.651·49-s − 0.778·51-s + 0.0602·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.099387335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099387335\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 6.68T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−10.16987167636969488936608876749, −9.465056027245491453115088468819, −8.531653632771439180155997871453, −7.45432129723567705892065173210, −6.49414200594878928316175807464, −5.74284660107487433367570605083, −5.15937037041393521223053614244, −3.76049263644581066220081406718, −2.62404841987271594434756390514, −0.889599051183806969294017096489,
0.889599051183806969294017096489, 2.62404841987271594434756390514, 3.76049263644581066220081406718, 5.15937037041393521223053614244, 5.74284660107487433367570605083, 6.49414200594878928316175807464, 7.45432129723567705892065173210, 8.531653632771439180155997871453, 9.465056027245491453115088468819, 10.16987167636969488936608876749
