| L(s) = 1 | + 1.51·3-s − 2.23·5-s + 3.60·7-s − 0.717·9-s + 1.50·11-s + 0.00459·13-s − 3.37·15-s + 1.11·17-s − 2.49·19-s + 5.44·21-s − 1.72·23-s − 0.00724·25-s − 5.61·27-s − 0.572·29-s − 3.25·31-s + 2.27·33-s − 8.05·35-s − 6.13·37-s + 0.00694·39-s − 5.89·41-s + 7.66·43-s + 1.60·45-s − 6.44·47-s + 5.98·49-s + 1.69·51-s − 7.82·53-s − 3.35·55-s + ⋯ |
| L(s) = 1 | + 0.872·3-s − 0.999·5-s + 1.36·7-s − 0.239·9-s + 0.453·11-s + 0.00127·13-s − 0.871·15-s + 0.271·17-s − 0.573·19-s + 1.18·21-s − 0.359·23-s − 0.00144·25-s − 1.08·27-s − 0.106·29-s − 0.584·31-s + 0.395·33-s − 1.36·35-s − 1.00·37-s + 0.00111·39-s − 0.920·41-s + 1.16·43-s + 0.239·45-s − 0.940·47-s + 0.854·49-s + 0.236·51-s − 1.07·53-s − 0.452·55-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 | \( 1 - 1.51T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 13 | \( 1 - 0.00459T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 + 0.572T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 + 5.89T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 + 7.82T + 53T^{2} \) |
| 59 | \( 1 + 0.253T + 59T^{2} \) |
| 61 | \( 1 - 7.08T + 61T^{2} \) |
| 67 | \( 1 + 6.16T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 1.99T + 79T^{2} \) |
| 83 | \( 1 + 6.91T + 83T^{2} \) |
| 89 | \( 1 - 3.37T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.65393421339527115881028391813, −7.11592885624991137059105681963, −6.08991868353795356816579626458, −5.26520357004341806976422661177, −4.49750155281297289629946743513, −3.84897562994749473276149429719, −3.23232395641563772444107442294, −2.18338612849222785981301334443, −1.47456512985157684613966121565, 0,
1.47456512985157684613966121565, 2.18338612849222785981301334443, 3.23232395641563772444107442294, 3.84897562994749473276149429719, 4.49750155281297289629946743513, 5.26520357004341806976422661177, 6.08991868353795356816579626458, 7.11592885624991137059105681963, 7.65393421339527115881028391813
