L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s − 2·13-s + 14-s + 16-s − 18-s + 21-s + 22-s + 23-s + 24-s + 2·26-s − 27-s − 28-s + 6·29-s + 8·31-s − 32-s + 33-s + 36-s − 3·37-s + 2·39-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 − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s − 0.493·37-s + 0.320·39-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
\(L(1)\) |
\(\approx\) |
\(0.5683867589\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5683867589\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.48831771889781, −12.11560317403345, −11.81915329609234, −11.23931097969159, −10.79135218183762, −10.25074144351872, −10.04941073387295, −9.585374082368026, −8.994244935870408, −8.541883398028296, −8.090064898902655, −7.525633700406067, −7.074939822503934, −6.684475771779887, −6.131586718181720, −5.728238164706924, −5.152524944045615, −4.526744873766388, −4.214447007901132, −3.320625871678033, −2.755608458284664, −2.459206328388414, −1.499842471547877, −1.061224095879830, −0.2614782855187145,
0.2614782855187145, 1.061224095879830, 1.499842471547877, 2.459206328388414, 2.755608458284664, 3.320625871678033, 4.214447007901132, 4.526744873766388, 5.152524944045615, 5.728238164706924, 6.131586718181720, 6.684475771779887, 7.074939822503934, 7.525633700406067, 8.090064898902655, 8.541883398028296, 8.994244935870408, 9.585374082368026, 10.04941073387295, 10.25074144351872, 10.79135218183762, 11.23931097969159, 11.81915329609234, 12.11560317403345, 12.48831771889781