| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s + 2·7-s + 9-s + 6·10-s − 5·11-s + 2·12-s − 4·14-s − 3·15-s − 4·16-s − 2·18-s + 5·19-s − 6·20-s + 2·21-s + 10·22-s + 23-s + 4·25-s + 27-s + 4·28-s + 6·29-s + 6·30-s + 10·31-s + 8·32-s − 5·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s + 0.755·7-s + 1/3·9-s + 1.89·10-s − 1.50·11-s + 0.577·12-s − 1.06·14-s − 0.774·15-s − 16-s − 0.471·18-s + 1.14·19-s − 1.34·20-s + 0.436·21-s + 2.13·22-s + 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 1.09·30-s + 1.79·31-s + 1.41·32-s − 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 146523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 146523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228199543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228199543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.43980819806782, −12.87083787477994, −12.12012253258997, −11.85298196943519, −11.34987971470492, −10.85022529273317, −10.36471368062152, −10.03814293479225, −9.496712078700621, −8.795335353886924, −8.368408110034223, −8.113453531601681, −7.737915760562117, −7.357386642538423, −6.878083194095051, −6.148553846588760, −5.143446747692702, −4.838126950188488, −4.380467105986074, −3.547367178562989, −2.993111088662713, −2.457596715913276, −1.758875870001984, −0.8958790128742061, −0.5281675866143078,
0.5281675866143078, 0.8958790128742061, 1.758875870001984, 2.457596715913276, 2.993111088662713, 3.547367178562989, 4.380467105986074, 4.838126950188488, 5.143446747692702, 6.148553846588760, 6.878083194095051, 7.357386642538423, 7.737915760562117, 8.113453531601681, 8.368408110034223, 8.795335353886924, 9.496712078700621, 10.03814293479225, 10.36471368062152, 10.85022529273317, 11.34987971470492, 11.85298196943519, 12.12012253258997, 12.87083787477994, 13.43980819806782
