| L(s) = 1 | + 3.04·3-s − 2.60·5-s + 6.26·9-s + 3.68·11-s + 6.17·13-s − 7.92·15-s − 6.92·17-s − 19-s − 2.61·23-s + 1.77·25-s + 9.94·27-s + 3.12·29-s + 6.97·31-s + 11.2·33-s + 0.292·37-s + 18.7·39-s + 5.29·41-s − 1.96·43-s − 16.3·45-s + 7.99·47-s − 21.0·51-s + 1.79·53-s − 9.58·55-s − 3.04·57-s + 7.12·59-s − 2.14·61-s − 16.0·65-s + ⋯ |
| L(s) = 1 | + 1.75·3-s − 1.16·5-s + 2.08·9-s + 1.11·11-s + 1.71·13-s − 2.04·15-s − 1.67·17-s − 0.229·19-s − 0.546·23-s + 0.354·25-s + 1.91·27-s + 0.579·29-s + 1.25·31-s + 1.95·33-s + 0.0480·37-s + 3.00·39-s + 0.826·41-s − 0.299·43-s − 2.43·45-s + 1.16·47-s − 2.95·51-s + 0.246·53-s − 1.29·55-s − 0.403·57-s + 0.927·59-s − 0.275·61-s − 1.99·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.784615852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.784615852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.04T + 3T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 - 6.17T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 6.97T + 31T^{2} \) |
| 37 | \( 1 - 0.292T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 - 7.99T + 47T^{2} \) |
| 53 | \( 1 - 1.79T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 7.63T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 7.35T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 8.90T + 97T^{2} \) |
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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
−8.187965599414505207824575884336, −7.35514371990263686575467550280, −6.68867316201325482733641443791, −6.10081378781812027440829794007, −4.49200446907709010609490758180, −4.08620130416854187966385880208, −3.67759925714835914760876808775, −2.81679607355224842747371682823, −1.94268461900725684328179818425, −0.937438010895845612405508485826,
0.937438010895845612405508485826, 1.94268461900725684328179818425, 2.81679607355224842747371682823, 3.67759925714835914760876808775, 4.08620130416854187966385880208, 4.49200446907709010609490758180, 6.10081378781812027440829794007, 6.68867316201325482733641443791, 7.35514371990263686575467550280, 8.187965599414505207824575884336
