| L(s) = 1 | + 1.47·3-s − 1.11·5-s − 7-s − 0.830·9-s + 2.22·11-s − 1.64·15-s − 17-s − 4.94·19-s − 1.47·21-s − 9.17·23-s − 3.75·25-s − 5.64·27-s + 0.945·29-s + 1.11·31-s + 3.28·33-s + 1.11·35-s − 4.94·37-s + 10.2·41-s + 11.2·43-s + 0.926·45-s − 9.89·47-s + 49-s − 1.47·51-s + 7.83·53-s − 2.48·55-s − 7.28·57-s − 13.7·59-s + ⋯ |
| L(s) = 1 | + 0.850·3-s − 0.498·5-s − 0.377·7-s − 0.276·9-s + 0.672·11-s − 0.423·15-s − 0.242·17-s − 1.13·19-s − 0.321·21-s − 1.91·23-s − 0.751·25-s − 1.08·27-s + 0.175·29-s + 0.200·31-s + 0.571·33-s + 0.188·35-s − 0.813·37-s + 1.60·41-s + 1.71·43-s + 0.138·45-s − 1.44·47-s + 0.142·49-s − 0.206·51-s + 1.07·53-s − 0.335·55-s − 0.964·57-s − 1.78·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 19 | \( 1 + 4.94T + 19T^{2} \) |
| 23 | \( 1 + 9.17T + 23T^{2} \) |
| 29 | \( 1 - 0.945T + 29T^{2} \) |
| 31 | \( 1 - 1.11T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 7.83T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 0.904T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 - 0.756T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.839605414514160320641722789277, −8.050094262371263042663552435426, −7.51542890178893285120226536209, −6.36164833087686256864331119335, −5.83623317887716347547937735355, −4.30450499563912716767404608256, −3.87652047629978531340788227141, −2.81276333203957859199325230635, −1.87599030954008853504029743434, 0,
1.87599030954008853504029743434, 2.81276333203957859199325230635, 3.87652047629978531340788227141, 4.30450499563912716767404608256, 5.83623317887716347547937735355, 6.36164833087686256864331119335, 7.51542890178893285120226536209, 8.050094262371263042663552435426, 8.839605414514160320641722789277
