| L(s) = 1 | + 2-s + 1.05·3-s + 4-s + 3.93·5-s + 1.05·6-s + 8-s − 1.88·9-s + 3.93·10-s + 0.653·11-s + 1.05·12-s + 13-s + 4.16·15-s + 16-s + 2.22·17-s − 1.88·18-s − 7.10·19-s + 3.93·20-s + 0.653·22-s + 4.11·23-s + 1.05·24-s + 10.5·25-s + 26-s − 5.16·27-s + 4.65·29-s + 4.16·30-s − 10.2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.610·3-s + 0.5·4-s + 1.76·5-s + 0.432·6-s + 0.353·8-s − 0.626·9-s + 1.24·10-s + 0.197·11-s + 0.305·12-s + 0.277·13-s + 1.07·15-s + 0.250·16-s + 0.539·17-s − 0.443·18-s − 1.63·19-s + 0.880·20-s + 0.139·22-s + 0.858·23-s + 0.216·24-s + 2.10·25-s + 0.196·26-s − 0.993·27-s + 0.864·29-s + 0.760·30-s − 1.83·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.038967285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.038967285\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.05T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 11 | \( 1 - 0.653T + 11T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 1.05T + 37T^{2} \) |
| 41 | \( 1 - 7.64T + 41T^{2} \) |
| 43 | \( 1 + 4.05T + 43T^{2} \) |
| 47 | \( 1 - 4.22T + 47T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 + 9.33T + 61T^{2} \) |
| 67 | \( 1 - 0.653T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 4.92T + 89T^{2} \) |
| 97 | \( 1 - 9.87T + 97T^{2} \) |
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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
−9.498064663002788860281109740917, −9.033983790616246790974467065154, −8.157919209995519082905289503058, −6.95851079072239206793185547482, −6.11907503644013923396911861501, −5.62677093964479478064693229699, −4.62660537052312071445355096592, −3.34935264894990761279379209894, −2.48525964061881990351310675720, −1.63588642154551308119113418374,
1.63588642154551308119113418374, 2.48525964061881990351310675720, 3.34935264894990761279379209894, 4.62660537052312071445355096592, 5.62677093964479478064693229699, 6.11907503644013923396911861501, 6.95851079072239206793185547482, 8.157919209995519082905289503058, 9.033983790616246790974467065154, 9.498064663002788860281109740917
