| L(s) = 1 | − 3-s + 4·5-s − 3·7-s − 2·9-s − 5·11-s − 13-s − 4·15-s − 8·19-s + 3·21-s + 9·23-s + 11·25-s + 5·27-s + 6·29-s − 2·31-s + 5·33-s − 12·35-s − 2·37-s + 39-s − 10·41-s − 5·43-s − 8·45-s − 47-s + 2·49-s + 6·53-s − 20·55-s + 8·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.13·7-s − 2/3·9-s − 1.50·11-s − 0.277·13-s − 1.03·15-s − 1.83·19-s + 0.654·21-s + 1.87·23-s + 11/5·25-s + 0.962·27-s + 1.11·29-s − 0.359·31-s + 0.870·33-s − 2.02·35-s − 0.328·37-s + 0.160·39-s − 1.56·41-s − 0.762·43-s − 1.19·45-s − 0.145·47-s + 2/7·49-s + 0.824·53-s − 2.69·55-s + 1.05·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8867862482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8867862482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−15.05464474214757, −14.56925037538656, −13.70250132704125, −13.43848798400471, −13.03157435319527, −12.53285524718638, −12.09703183280892, −10.95846142344012, −10.76725958434327, −10.22380091830172, −9.894336693479092, −9.092662208864203, −8.743062276812244, −8.129882592372276, −7.053771061889386, −6.557282317085260, −6.288455727085744, −5.575238104517362, −5.093776445678250, −4.741571344148264, −3.391691439554144, −2.734296721786162, −2.454887162921314, −1.497835193164699, −0.3525466672894680,
0.3525466672894680, 1.497835193164699, 2.454887162921314, 2.734296721786162, 3.391691439554144, 4.741571344148264, 5.093776445678250, 5.575238104517362, 6.288455727085744, 6.557282317085260, 7.053771061889386, 8.129882592372276, 8.743062276812244, 9.092662208864203, 9.894336693479092, 10.22380091830172, 10.76725958434327, 10.95846142344012, 12.09703183280892, 12.53285524718638, 13.03157435319527, 13.43848798400471, 13.70250132704125, 14.56925037538656, 15.05464474214757
