| L(s) = 1 | + 0.527·2-s − 1.24·3-s − 1.72·4-s − 0.656·6-s − 3.22·7-s − 1.96·8-s − 1.45·9-s + 11-s + 2.14·12-s + 4.87·13-s − 1.69·14-s + 2.41·16-s − 7.11·17-s − 0.764·18-s + 19-s + 4.01·21-s + 0.527·22-s − 1.12·23-s + 2.44·24-s + 2.56·26-s + 5.53·27-s + 5.54·28-s + 8.47·29-s − 0.926·31-s + 5.19·32-s − 1.24·33-s − 3.75·34-s + ⋯ |
| L(s) = 1 | + 0.372·2-s − 0.718·3-s − 0.861·4-s − 0.267·6-s − 1.21·7-s − 0.693·8-s − 0.483·9-s + 0.301·11-s + 0.618·12-s + 1.35·13-s − 0.453·14-s + 0.602·16-s − 1.72·17-s − 0.180·18-s + 0.229·19-s + 0.875·21-s + 0.112·22-s − 0.234·23-s + 0.498·24-s + 0.503·26-s + 1.06·27-s + 1.04·28-s + 1.57·29-s − 0.166·31-s + 0.918·32-s − 0.216·33-s − 0.643·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.527T + 2T^{2} \) |
| 3 | \( 1 + 1.24T + 3T^{2} \) |
| 7 | \( 1 + 3.22T + 7T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 0.926T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.49T + 41T^{2} \) |
| 43 | \( 1 - 7.22T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 7.92T + 53T^{2} \) |
| 59 | \( 1 - 1.91T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 0.979T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 3.05T + 79T^{2} \) |
| 83 | \( 1 - 2.60T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.008673166298826607309048462151, −6.69387032436688384336911782187, −6.16174379399124792406094367991, −5.96899685855999758001447152420, −4.78176790486215781689974225560, −4.28014787743017484488657647331, −3.38302069549451511399520385991, −2.67707971515479932748775502007, −0.981885962437332969368780517883, 0,
0.981885962437332969368780517883, 2.67707971515479932748775502007, 3.38302069549451511399520385991, 4.28014787743017484488657647331, 4.78176790486215781689974225560, 5.96899685855999758001447152420, 6.16174379399124792406094367991, 6.69387032436688384336911782187, 8.008673166298826607309048462151
