| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 9-s + 6·11-s − 2·13-s − 4·15-s − 4·21-s + 6·23-s − 25-s − 4·27-s − 10·29-s + 2·31-s + 12·33-s + 4·35-s + 6·37-s − 4·39-s + 6·41-s − 8·43-s − 2·45-s − 3·49-s + 10·53-s − 12·55-s − 8·59-s + 14·61-s − 2·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 1.03·15-s − 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.769·27-s − 1.85·29-s + 0.359·31-s + 2.08·33-s + 0.676·35-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 3/7·49-s + 1.37·53-s − 1.61·55-s − 1.04·59-s + 1.79·61-s − 0.251·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.382501879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382501879\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.45436845429851, −15.19293076009787, −14.64895418176290, −14.33589441909798, −13.61659540093980, −13.01764459540227, −12.61623312882893, −11.75902872687582, −11.49872801690276, −10.90629740166114, −9.743625103939916, −9.554576153322534, −9.045318264607327, −8.449398092470946, −7.849321855615433, −7.201216849072458, −6.786942117561365, −6.034547702671904, −5.192319897555826, −4.157401531637930, −3.834060753775994, −3.281359471836151, −2.560516067500682, −1.679486256750838, −0.6044083914825930,
0.6044083914825930, 1.679486256750838, 2.560516067500682, 3.281359471836151, 3.834060753775994, 4.157401531637930, 5.192319897555826, 6.034547702671904, 6.786942117561365, 7.201216849072458, 7.849321855615433, 8.449398092470946, 9.045318264607327, 9.554576153322534, 9.743625103939916, 10.90629740166114, 11.49872801690276, 11.75902872687582, 12.61623312882893, 13.01764459540227, 13.61659540093980, 14.33589441909798, 14.64895418176290, 15.19293076009787, 15.45436845429851
