| L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s − 9·5-s + 4·6-s − 31·7-s + 8·8-s − 23·9-s − 18·10-s + 57·11-s + 8·12-s + 52·13-s − 62·14-s − 18·15-s + 16·16-s + 69·17-s − 46·18-s − 36·20-s − 62·21-s + 114·22-s − 72·23-s + 16·24-s − 44·25-s + 104·26-s − 100·27-s − 124·28-s + 150·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.804·5-s + 0.272·6-s − 1.67·7-s + 0.353·8-s − 0.851·9-s − 0.569·10-s + 1.56·11-s + 0.192·12-s + 1.10·13-s − 1.18·14-s − 0.309·15-s + 1/4·16-s + 0.984·17-s − 0.602·18-s − 0.402·20-s − 0.644·21-s + 1.10·22-s − 0.652·23-s + 0.136·24-s − 0.351·25-s + 0.784·26-s − 0.712·27-s − 0.836·28-s + 0.960·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.707634957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.707634957\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 31 T + p^{3} T^{2} \) |
| 11 | \( 1 - 57 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 69 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 226 T + p^{3} T^{2} \) |
| 41 | \( 1 - 258 T + p^{3} T^{2} \) |
| 43 | \( 1 + 67 T + p^{3} T^{2} \) |
| 47 | \( 1 - 579 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 330 T + p^{3} T^{2} \) |
| 61 | \( 1 + 13 T + p^{3} T^{2} \) |
| 67 | \( 1 - 856 T + p^{3} T^{2} \) |
| 71 | \( 1 + 642 T + p^{3} T^{2} \) |
| 73 | \( 1 + 487 T + p^{3} T^{2} \) |
| 79 | \( 1 - 700 T + p^{3} T^{2} \) |
| 83 | \( 1 + 12 T + p^{3} T^{2} \) |
| 89 | \( 1 - 600 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1424 T + p^{3} T^{2} \) |
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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
−9.943545948746466181012360584821, −9.134022269241099826866443786294, −8.309510880483921960563472539472, −7.24336399279585352215141147687, −6.26406550716386422885545682750, −5.82325973600734461519755442243, −3.92572618536794145210362457236, −3.72132606505921777833389911323, −2.69611644098660939672203613610, −0.835463967705520726607719233447,
0.835463967705520726607719233447, 2.69611644098660939672203613610, 3.72132606505921777833389911323, 3.92572618536794145210362457236, 5.82325973600734461519755442243, 6.26406550716386422885545682750, 7.24336399279585352215141147687, 8.309510880483921960563472539472, 9.134022269241099826866443786294, 9.943545948746466181012360584821
