| L(s) = 1 | − 3-s + 9-s − 6·11-s − 6·13-s − 4·19-s − 27-s + 8·29-s − 2·31-s + 6·33-s + 4·37-s + 6·39-s + 10·41-s − 6·43-s − 2·47-s + 10·53-s + 4·57-s − 4·59-s + 14·61-s + 14·67-s + 8·71-s + 6·73-s − 8·79-s + 81-s + 8·83-s − 8·87-s + 18·89-s + 2·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.66·13-s − 0.917·19-s − 0.192·27-s + 1.48·29-s − 0.359·31-s + 1.04·33-s + 0.657·37-s + 0.960·39-s + 1.56·41-s − 0.914·43-s − 0.291·47-s + 1.37·53-s + 0.529·57-s − 0.520·59-s + 1.79·61-s + 1.71·67-s + 0.949·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s + 0.878·83-s − 0.857·87-s + 1.90·89-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092018999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092018999\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.73709471223746, −12.57836493757136, −12.01763135520206, −11.57620274183138, −10.88766453842518, −10.66016044132689, −10.15916721476592, −9.760502878433914, −9.376924533696371, −8.521490845732536, −8.129022978141035, −7.762277339223045, −7.175722289113088, −6.760536757709449, −6.243255936415926, −5.529899011825237, −5.182841539273054, −4.815020818862975, −4.286295327480628, −3.634977014111589, −2.715167104126807, −2.473344993326732, −2.022894970289309, −0.8693677974528036, −0.3681981112272259,
0.3681981112272259, 0.8693677974528036, 2.022894970289309, 2.473344993326732, 2.715167104126807, 3.634977014111589, 4.286295327480628, 4.815020818862975, 5.182841539273054, 5.529899011825237, 6.243255936415926, 6.760536757709449, 7.175722289113088, 7.762277339223045, 8.129022978141035, 8.521490845732536, 9.376924533696371, 9.760502878433914, 10.15916721476592, 10.66016044132689, 10.88766453842518, 11.57620274183138, 12.01763135520206, 12.57836493757136, 12.73709471223746
