| L(s) = 1 | + 2-s − 0.732·3-s + 4-s + 5-s − 0.732·6-s − 7-s + 8-s − 2.46·9-s + 10-s − 0.732·12-s − 5.46·13-s − 14-s − 0.732·15-s + 16-s + 3.46·17-s − 2.46·18-s − 0.732·19-s + 20-s + 0.732·21-s + 4.73·23-s − 0.732·24-s + 25-s − 5.46·26-s + 4·27-s − 28-s + 1.26·29-s − 0.732·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.422·3-s + 0.5·4-s + 0.447·5-s − 0.298·6-s − 0.377·7-s + 0.353·8-s − 0.821·9-s + 0.316·10-s − 0.211·12-s − 1.51·13-s − 0.267·14-s − 0.189·15-s + 0.250·16-s + 0.840·17-s − 0.580·18-s − 0.167·19-s + 0.223·20-s + 0.159·21-s + 0.986·23-s − 0.149·24-s + 0.200·25-s − 1.07·26-s + 0.769·27-s − 0.188·28-s + 0.235·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.301690419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301690419\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 0.732T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 1.26T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.51602062909253101135901703858, −7.00521354998952660543979866252, −6.26015855164109240790167232095, −5.59995393517683811258942768870, −5.13176064360016745125757565234, −4.46786232377923946802825920134, −3.33941431028547672236942299646, −2.80843255119649005781846290588, −1.98340969892397530451168068999, −0.65572315744355252589267526058,
0.65572315744355252589267526058, 1.98340969892397530451168068999, 2.80843255119649005781846290588, 3.33941431028547672236942299646, 4.46786232377923946802825920134, 5.13176064360016745125757565234, 5.59995393517683811258942768870, 6.26015855164109240790167232095, 7.00521354998952660543979866252, 7.51602062909253101135901703858
