| L(s) = 1 | + 2-s + 2.73·3-s + 4-s − 5-s + 2.73·6-s − 7-s + 8-s + 4.46·9-s − 10-s + 2.73·12-s − 4·13-s − 14-s − 2.73·15-s + 16-s − 0.535·17-s + 4.46·18-s + 0.732·19-s − 20-s − 2.73·21-s + 8.19·23-s + 2.73·24-s + 25-s − 4·26-s + 3.99·27-s − 28-s + 7.66·29-s − 2.73·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.447·5-s + 1.11·6-s − 0.377·7-s + 0.353·8-s + 1.48·9-s − 0.316·10-s + 0.788·12-s − 1.10·13-s − 0.267·14-s − 0.705·15-s + 0.250·16-s − 0.129·17-s + 1.05·18-s + 0.167·19-s − 0.223·20-s − 0.596·21-s + 1.70·23-s + 0.557·24-s + 0.200·25-s − 0.784·26-s + 0.769·27-s − 0.188·28-s + 1.42·29-s − 0.498·30-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
| \(L(1)\) |
\(\approx\) |
\(5.464001302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.464001302\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 3T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 - 7.66T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 5.26T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 7.66T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.92T + 89T^{2} \) |
| 97 | \( 1 - 2.73T + 97T^{2} \) |
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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
−7.70014702013236625185555076843, −7.07283322824950512924685272291, −6.77277104262888667624762237853, −5.52023223740105866872014185815, −4.81341498321746847261400073540, −4.10031726532696832066542211065, −3.40313782158905782420318751953, −2.73108234649569775123328808523, −2.29668255169620977761687983713, −0.959330441740456959268937777453,
0.959330441740456959268937777453, 2.29668255169620977761687983713, 2.73108234649569775123328808523, 3.40313782158905782420318751953, 4.10031726532696832066542211065, 4.81341498321746847261400073540, 5.52023223740105866872014185815, 6.77277104262888667624762237853, 7.07283322824950512924685272291, 7.70014702013236625185555076843
