| L(s) = 1 | − 1.41·3-s − 2.82·5-s − 0.999·9-s − 2·11-s + 4.00·15-s − 1.41·17-s − 7.07·19-s + 4·23-s + 3.00·25-s + 5.65·27-s − 2·29-s − 8.48·31-s + 2.82·33-s − 10·37-s − 9.89·41-s + 2·43-s + 2.82·45-s − 2.82·47-s + 2.00·51-s + 2·53-s + 5.65·55-s + 10.0·57-s − 1.41·59-s − 2.82·61-s + 12·67-s − 5.65·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 1.26·5-s − 0.333·9-s − 0.603·11-s + 1.03·15-s − 0.342·17-s − 1.62·19-s + 0.834·23-s + 0.600·25-s + 1.08·27-s − 0.371·29-s − 1.52·31-s + 0.492·33-s − 1.64·37-s − 1.54·41-s + 0.304·43-s + 0.421·45-s − 0.412·47-s + 0.280·51-s + 0.274·53-s + 0.762·55-s + 1.32·57-s − 0.184·59-s − 0.362·61-s + 1.46·67-s − 0.681·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3003398721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3003398721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 + 7.07T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−8.510015737391016441751130816610, −8.042524361855270348276077393488, −7.03754578731282198030165533125, −6.58656412064968387452214377441, −5.47776322684104849845502864634, −4.95414257832187937683493415845, −4.00401407283564639225576083628, −3.25331568384605900256693060298, −2.01710438296605247614068955043, −0.32602621176531306770886831326,
0.32602621176531306770886831326, 2.01710438296605247614068955043, 3.25331568384605900256693060298, 4.00401407283564639225576083628, 4.95414257832187937683493415845, 5.47776322684104849845502864634, 6.58656412064968387452214377441, 7.03754578731282198030165533125, 8.042524361855270348276077393488, 8.510015737391016441751130816610
