| L(s) = 1 | + 3.02·3-s − 1.92·5-s − 4.18·7-s + 6.16·9-s + 1.99·11-s + 1.74·13-s − 5.82·15-s − 2.18·17-s + 1.78·19-s − 12.6·21-s + 3.66·23-s − 1.29·25-s + 9.58·27-s − 4.48·29-s − 10.3·31-s + 6.05·33-s + 8.05·35-s + 1.81·37-s + 5.28·39-s − 5.92·41-s − 3.30·43-s − 11.8·45-s − 0.751·47-s + 10.5·49-s − 6.60·51-s − 1.29·53-s − 3.84·55-s + ⋯ |
| L(s) = 1 | + 1.74·3-s − 0.860·5-s − 1.58·7-s + 2.05·9-s + 0.602·11-s + 0.484·13-s − 1.50·15-s − 0.529·17-s + 0.408·19-s − 2.76·21-s + 0.764·23-s − 0.259·25-s + 1.84·27-s − 0.832·29-s − 1.85·31-s + 1.05·33-s + 1.36·35-s + 0.297·37-s + 0.846·39-s − 0.925·41-s − 0.503·43-s − 1.76·45-s − 0.109·47-s + 1.50·49-s − 0.925·51-s − 0.178·53-s − 0.518·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.02T + 3T^{2} \) |
| 5 | \( 1 + 1.92T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 2.18T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 + 5.92T + 41T^{2} \) |
| 43 | \( 1 + 3.30T + 43T^{2} \) |
| 47 | \( 1 + 0.751T + 47T^{2} \) |
| 53 | \( 1 + 1.29T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 + 0.765T + 61T^{2} \) |
| 67 | \( 1 - 0.202T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 0.211T + 79T^{2} \) |
| 83 | \( 1 - 9.33T + 83T^{2} \) |
| 89 | \( 1 - 0.912T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−7.54632393284370257440016124982, −6.96511070946395732217334632217, −6.44175608069209232679497563708, −5.34599345068758358431213199958, −4.14452805745429540690567255855, −3.63686273515912182316328744674, −3.33527488697075737048864264247, −2.47140217141521147729406677066, −1.46436350369554941712900149375, 0,
1.46436350369554941712900149375, 2.47140217141521147729406677066, 3.33527488697075737048864264247, 3.63686273515912182316328744674, 4.14452805745429540690567255855, 5.34599345068758358431213199958, 6.44175608069209232679497563708, 6.96511070946395732217334632217, 7.54632393284370257440016124982
