| L(s) = 1 | − 3.39·3-s + 3.96·7-s + 8.54·9-s + 3.01·11-s + 4.83·13-s − 17-s + 6.08·19-s − 13.4·21-s − 7.81·23-s − 18.8·27-s + 8.34·29-s − 1.76·31-s − 10.2·33-s + 2.08·37-s − 16.4·39-s − 0.710·41-s + 1.54·43-s − 4.25·47-s + 8.70·49-s + 3.39·51-s − 0.584·53-s − 20.6·57-s − 2.34·59-s − 2.79·61-s + 33.8·63-s + 13.0·67-s + 26.5·69-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 1.49·7-s + 2.84·9-s + 0.910·11-s + 1.34·13-s − 0.242·17-s + 1.39·19-s − 2.93·21-s − 1.62·23-s − 3.62·27-s + 1.54·29-s − 0.317·31-s − 1.78·33-s + 0.342·37-s − 2.63·39-s − 0.111·41-s + 0.235·43-s − 0.620·47-s + 1.24·49-s + 0.475·51-s − 0.0803·53-s − 2.73·57-s − 0.304·59-s − 0.357·61-s + 4.26·63-s + 1.59·67-s + 3.19·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.291269984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.291269984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 3.39T + 3T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 19 | \( 1 - 6.08T + 19T^{2} \) |
| 23 | \( 1 + 7.81T + 23T^{2} \) |
| 29 | \( 1 - 8.34T + 29T^{2} \) |
| 31 | \( 1 + 1.76T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 + 0.710T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 + 4.25T + 47T^{2} \) |
| 53 | \( 1 + 0.584T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 1.66T + 71T^{2} \) |
| 73 | \( 1 - 8.12T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 - 7.16T + 97T^{2} \) |
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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
−9.578326760972000421632033497475, −8.415870087660189774223048374581, −7.64410587233343206203907397624, −6.67920579821275844983356604014, −6.04841033666370440251747404017, −5.32658489976058161059836150604, −4.54655920039010842624090995747, −3.85304753976495509827279837032, −1.65933036548477905243307779754, −0.989619336458783456183100046123,
0.989619336458783456183100046123, 1.65933036548477905243307779754, 3.85304753976495509827279837032, 4.54655920039010842624090995747, 5.32658489976058161059836150604, 6.04841033666370440251747404017, 6.67920579821275844983356604014, 7.64410587233343206203907397624, 8.415870087660189774223048374581, 9.578326760972000421632033497475
