| L(s) = 1 | + 1.48·3-s + 1.91·5-s + 4.50·7-s − 0.798·9-s + 4.79·11-s + 0.113·13-s + 2.84·15-s − 17-s − 3.79·19-s + 6.67·21-s + 2.66·23-s − 1.33·25-s − 5.63·27-s − 8.92·29-s + 6.56·31-s + 7.11·33-s + 8.61·35-s + 6.67·37-s + 0.168·39-s − 4.36·41-s + 4.49·43-s − 1.52·45-s + 7.24·47-s + 13.2·49-s − 1.48·51-s − 5.88·53-s + 9.18·55-s + ⋯ |
| L(s) = 1 | + 0.856·3-s + 0.856·5-s + 1.70·7-s − 0.266·9-s + 1.44·11-s + 0.0314·13-s + 0.733·15-s − 0.242·17-s − 0.870·19-s + 1.45·21-s + 0.556·23-s − 0.266·25-s − 1.08·27-s − 1.65·29-s + 1.17·31-s + 1.23·33-s + 1.45·35-s + 1.09·37-s + 0.0269·39-s − 0.682·41-s + 0.685·43-s − 0.227·45-s + 1.05·47-s + 1.89·49-s − 0.207·51-s − 0.808·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.516454687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.516454687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.48T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.113T + 13T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 6.67T + 37T^{2} \) |
| 41 | \( 1 + 4.36T + 41T^{2} \) |
| 43 | \( 1 - 4.49T + 43T^{2} \) |
| 47 | \( 1 - 7.24T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 - 8.13T + 79T^{2} \) |
| 83 | \( 1 - 4.89T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−7.976724560196999095866979945216, −7.30687593853613758738240664970, −6.36004410127020244464973253929, −5.81914722258555841760574747438, −4.96578450336070924303890875595, −4.24374794514005030524157648608, −3.57148755655376872227320927701, −2.33703263960607634682228178993, −1.97582444807080173440565781752, −1.09013437150364763411195581140,
1.09013437150364763411195581140, 1.97582444807080173440565781752, 2.33703263960607634682228178993, 3.57148755655376872227320927701, 4.24374794514005030524157648608, 4.96578450336070924303890875595, 5.81914722258555841760574747438, 6.36004410127020244464973253929, 7.30687593853613758738240664970, 7.976724560196999095866979945216
