| L(s) = 1 | + 2-s + 4-s − 3.28·5-s − 2.81·7-s + 8-s − 3.28·10-s − 5.17·11-s + 13-s − 2.81·14-s + 16-s + 0.699·17-s + 7.17·19-s − 3.28·20-s − 5.17·22-s + 6·23-s + 5.81·25-s + 26-s − 2.81·28-s + 2.22·29-s + 2.77·31-s + 32-s + 0.699·34-s + 9.24·35-s − 0.712·37-s + 7.17·38-s − 3.28·40-s + 0.823·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.47·5-s − 1.06·7-s + 0.353·8-s − 1.03·10-s − 1.56·11-s + 0.277·13-s − 0.751·14-s + 0.250·16-s + 0.169·17-s + 1.64·19-s − 0.735·20-s − 1.10·22-s + 1.25·23-s + 1.16·25-s + 0.196·26-s − 0.531·28-s + 0.412·29-s + 0.498·31-s + 0.176·32-s + 0.119·34-s + 1.56·35-s − 0.117·37-s + 1.16·38-s − 0.519·40-s + 0.128·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580452584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580452584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3.28T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 17 | \( 1 - 0.699T + 17T^{2} \) |
| 19 | \( 1 - 7.17T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2.22T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 0.712T + 37T^{2} \) |
| 41 | \( 1 - 0.823T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 - 0.445T + 67T^{2} \) |
| 71 | \( 1 - 1.18T + 71T^{2} \) |
| 73 | \( 1 + 0.222T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 7.97T + 83T^{2} \) |
| 89 | \( 1 + 7.39T + 89T^{2} \) |
| 97 | \( 1 - 4.95T + 97T^{2} \) |
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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
−9.049644177755802915618331924758, −8.048978219848994198142449597047, −7.47564453668846714107084875962, −6.89957202334116673744284503935, −5.77758984271584934451587791298, −5.05452188054136601679141898462, −4.14387883647368149406040672748, −3.18324723933712891191900865568, −2.83026902598674763433937056427, −0.72619474160192489141459170160,
0.72619474160192489141459170160, 2.83026902598674763433937056427, 3.18324723933712891191900865568, 4.14387883647368149406040672748, 5.05452188054136601679141898462, 5.77758984271584934451587791298, 6.89957202334116673744284503935, 7.47564453668846714107084875962, 8.048978219848994198142449597047, 9.049644177755802915618331924758
