L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s + 14-s + 16-s − 18-s + 21-s − 3·22-s + 4·23-s + 24-s − 27-s − 28-s + 3·29-s + 5·31-s − 32-s − 3·33-s + 36-s + 4·37-s − 42-s − 6·43-s + 3·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.898·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + 0.657·37-s − 0.154·42-s − 0.914·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.90337335287741, −12.21034660667604, −11.96738208118487, −11.45587428861119, −11.12390139969581, −10.62459372072234, −10.04148530697042, −9.704446968792022, −9.418164232294542, −8.637210759768047, −8.457417285395852, −7.847541284722488, −7.237516231121828, −6.712678277865739, −6.516038486347201, −6.072068388837486, −5.271312286835857, −5.010914576938852, −4.240995251189096, −3.781067552699453, −3.141729287981186, −2.602494659009291, −1.914217179657744, −1.185843583587011, −0.8227542818965131, 0,
0.8227542818965131, 1.185843583587011, 1.914217179657744, 2.602494659009291, 3.141729287981186, 3.781067552699453, 4.240995251189096, 5.010914576938852, 5.271312286835857, 6.072068388837486, 6.516038486347201, 6.712678277865739, 7.237516231121828, 7.847541284722488, 8.457417285395852, 8.637210759768047, 9.418164232294542, 9.704446968792022, 10.04148530697042, 10.62459372072234, 11.12390139969581, 11.45587428861119, 11.96738208118487, 12.21034660667604, 12.90337335287741