| L(s) = 1 | + 2-s + 3.37·3-s + 4-s + 3.37·6-s − 3.37·7-s + 8-s + 8.37·9-s − 11-s + 3.37·12-s − 2·13-s − 3.37·14-s + 16-s − 1.37·17-s + 8.37·18-s + 0.627·19-s − 11.3·21-s − 22-s − 2.74·23-s + 3.37·24-s − 2·26-s + 18.1·27-s − 3.37·28-s + 1.37·29-s + 3.37·31-s + 32-s − 3.37·33-s − 1.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.94·3-s + 0.5·4-s + 1.37·6-s − 1.27·7-s + 0.353·8-s + 2.79·9-s − 0.301·11-s + 0.973·12-s − 0.554·13-s − 0.901·14-s + 0.250·16-s − 0.332·17-s + 1.97·18-s + 0.144·19-s − 2.48·21-s − 0.213·22-s − 0.572·23-s + 0.688·24-s − 0.392·26-s + 3.48·27-s − 0.637·28-s + 0.254·29-s + 0.605·31-s + 0.176·32-s − 0.587·33-s − 0.235·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.563213580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.563213580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.37T + 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 0.627T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 5.37T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 - 2.74T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.41212137948225363058205168160, −9.864761795108724706025420939972, −9.020387160614139972502331625973, −8.127424455272814834874552417123, −7.19727961867243120122545681911, −6.47217694077729236639419910749, −4.85996959822379482952895945465, −3.70635389086485072869558292536, −3.05411219239337642674906067342, −2.07598733130289166419617282195,
2.07598733130289166419617282195, 3.05411219239337642674906067342, 3.70635389086485072869558292536, 4.85996959822379482952895945465, 6.47217694077729236639419910749, 7.19727961867243120122545681911, 8.127424455272814834874552417123, 9.020387160614139972502331625973, 9.864761795108724706025420939972, 10.41212137948225363058205168160
