L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s + 2·13-s + 4·14-s − 2·15-s − 16-s − 2·17-s + 18-s − 2·20-s − 4·21-s − 22-s − 8·23-s + 3·24-s − 25-s + 2·26-s − 27-s − 4·28-s + 6·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.447·20-s − 0.872·21-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.603113292\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.603113292\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 47 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.01664064445573, −13.70270520345324, −13.25875300717129, −12.61100756576642, −12.13203496359075, −11.61764615858015, −11.29333178443116, −10.58355868430938, −10.06963715892632, −9.724658641102546, −8.989868642678213, −8.313592810535341, −8.166393830908442, −7.440520162594419, −6.460728632517970, −6.143118470703632, −5.733576812887000, −5.022756828682224, −4.747656124846562, −4.179206189975345, −3.641210576130372, −2.522623642993669, −2.183198931434465, −1.287443851725668, −0.6220188627890586,
0.6220188627890586, 1.287443851725668, 2.183198931434465, 2.522623642993669, 3.641210576130372, 4.179206189975345, 4.747656124846562, 5.022756828682224, 5.733576812887000, 6.143118470703632, 6.460728632517970, 7.440520162594419, 8.166393830908442, 8.313592810535341, 8.989868642678213, 9.724658641102546, 10.06963715892632, 10.58355868430938, 11.29333178443116, 11.61764615858015, 12.13203496359075, 12.61100756576642, 13.25875300717129, 13.70270520345324, 14.01664064445573