| L(s) = 1 | − 3.18·3-s + 3.42·5-s + 7.12·9-s − 11-s − 3.66·13-s − 10.8·15-s + 1.76·17-s − 3.76·19-s − 1.23·23-s + 6.70·25-s − 13.1·27-s − 2.89·29-s − 9.46·31-s + 3.18·33-s + 2.84·37-s + 11.6·39-s − 8.70·41-s + 12.0·43-s + 24.3·45-s − 4.23·47-s − 5.60·51-s − 0.225·53-s − 3.42·55-s + 11.9·57-s + 6.54·59-s + 10.2·61-s − 12.5·65-s + ⋯ |
| L(s) = 1 | − 1.83·3-s + 1.52·5-s + 2.37·9-s − 0.301·11-s − 1.01·13-s − 2.81·15-s + 0.427·17-s − 0.862·19-s − 0.258·23-s + 1.34·25-s − 2.52·27-s − 0.538·29-s − 1.69·31-s + 0.553·33-s + 0.467·37-s + 1.86·39-s − 1.35·41-s + 1.84·43-s + 3.63·45-s − 0.618·47-s − 0.784·51-s − 0.0309·53-s − 0.461·55-s + 1.58·57-s + 0.852·59-s + 1.31·61-s − 1.55·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 13 | \( 1 + 3.66T + 13T^{2} \) |
| 17 | \( 1 - 1.76T + 17T^{2} \) |
| 19 | \( 1 + 3.76T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 2.84T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 - 6.54T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 + 4.23T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 7.26T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.970844138618085141130067584004, −7.58252170731031939679881216304, −6.87298511591025444037528648770, −6.13392634668717030402835857289, −5.50222773928328499936042188261, −5.11013443903914427575740094896, −4.07098934202113921363975679377, −2.38272790730089366221582184887, −1.47279979604657667075187088065, 0,
1.47279979604657667075187088065, 2.38272790730089366221582184887, 4.07098934202113921363975679377, 5.11013443903914427575740094896, 5.50222773928328499936042188261, 6.13392634668717030402835857289, 6.87298511591025444037528648770, 7.58252170731031939679881216304, 8.970844138618085141130067584004
