| L(s) = 1 | − 2.41·3-s + 1.82·5-s + 4.41·7-s + 2.82·9-s + 0.828·11-s − 13-s − 4.41·15-s + 17-s + 5.65·19-s − 10.6·21-s + 8.82·23-s − 1.65·25-s + 0.414·27-s − 3.65·29-s + 3.65·31-s − 1.99·33-s + 8.07·35-s − 7·37-s + 2.41·39-s + 9.65·41-s + 8.41·43-s + 5.17·45-s + 0.757·47-s + 12.4·49-s − 2.41·51-s − 3.65·53-s + 1.51·55-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.817·5-s + 1.66·7-s + 0.942·9-s + 0.249·11-s − 0.277·13-s − 1.13·15-s + 0.242·17-s + 1.29·19-s − 2.32·21-s + 1.84·23-s − 0.331·25-s + 0.0797·27-s − 0.679·29-s + 0.656·31-s − 0.348·33-s + 1.36·35-s − 1.15·37-s + 0.386·39-s + 1.50·41-s + 1.28·43-s + 0.770·45-s + 0.110·47-s + 1.78·49-s − 0.338·51-s − 0.502·53-s + 0.204·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.830938061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830938061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 9.65T + 41T^{2} \) |
| 43 | \( 1 - 8.41T + 43T^{2} \) |
| 47 | \( 1 - 0.757T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.758064539530912324148640529626, −7.56629676593892581009407910043, −7.24203978358137440590505313893, −6.06916666069185765720303208900, −5.58234990639318565746636160697, −4.98007214874741965981735348111, −4.40457940797217116612463190000, −2.92293303754370579272386865359, −1.66029879489588885104772795668, −0.963694288477816318568717285644,
0.963694288477816318568717285644, 1.66029879489588885104772795668, 2.92293303754370579272386865359, 4.40457940797217116612463190000, 4.98007214874741965981735348111, 5.58234990639318565746636160697, 6.06916666069185765720303208900, 7.24203978358137440590505313893, 7.56629676593892581009407910043, 8.758064539530912324148640529626
