| L(s) = 1 | + 1.33·2-s − 0.228·4-s − 3.76·7-s − 2.96·8-s − 5.03·11-s − 4.92·13-s − 5.00·14-s − 3.48·16-s − 7.40·17-s − 0.692·19-s − 6.70·22-s + 4.29·23-s − 6.55·26-s + 0.861·28-s − 4.13·29-s − 3.81·31-s + 1.28·32-s − 9.85·34-s + 37-s − 0.921·38-s + 1.17·41-s + 1.22·43-s + 1.15·44-s + 5.71·46-s − 5.14·47-s + 7.16·49-s + 1.12·52-s + ⋯ |
| L(s) = 1 | + 0.941·2-s − 0.114·4-s − 1.42·7-s − 1.04·8-s − 1.51·11-s − 1.36·13-s − 1.33·14-s − 0.872·16-s − 1.79·17-s − 0.158·19-s − 1.42·22-s + 0.895·23-s − 1.28·26-s + 0.162·28-s − 0.768·29-s − 0.685·31-s + 0.227·32-s − 1.68·34-s + 0.164·37-s − 0.149·38-s + 0.184·41-s + 0.186·43-s + 0.173·44-s + 0.842·46-s − 0.750·47-s + 1.02·49-s + 0.156·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1913694897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1913694897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 + 7.40T + 17T^{2} \) |
| 19 | \( 1 + 0.692T + 19T^{2} \) |
| 23 | \( 1 - 4.29T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 3.81T + 31T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 1.22T + 43T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 + 8.01T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 3.94T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.54601852202271254692630937046, −7.00645271745604497328937830888, −6.29889971602807502497982489785, −5.60468968419167449796226071985, −4.94390527635192270819563552139, −4.40296456603570652168292371985, −3.48997986992738004933561450388, −2.75278411894572612052229390347, −2.31119280414094961188173722244, −0.16773104664361240372135157248,
0.16773104664361240372135157248, 2.31119280414094961188173722244, 2.75278411894572612052229390347, 3.48997986992738004933561450388, 4.40296456603570652168292371985, 4.94390527635192270819563552139, 5.60468968419167449796226071985, 6.29889971602807502497982489785, 7.00645271745604497328937830888, 7.54601852202271254692630937046
