| L(s) = 1 | + 2-s + 3·3-s − 4-s − 3·5-s + 3·6-s − 3·8-s + 6·9-s − 3·10-s − 3·11-s − 3·12-s − 9·15-s − 16-s + 2·17-s + 6·18-s + 19-s + 3·20-s − 3·22-s − 9·24-s + 4·25-s + 9·27-s + 7·29-s − 9·30-s − 3·31-s + 5·32-s − 9·33-s + 2·34-s − 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s − 1.34·5-s + 1.22·6-s − 1.06·8-s + 2·9-s − 0.948·10-s − 0.904·11-s − 0.866·12-s − 2.32·15-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.229·19-s + 0.670·20-s − 0.639·22-s − 1.83·24-s + 4/5·25-s + 1.73·27-s + 1.29·29-s − 1.64·30-s − 0.538·31-s + 0.883·32-s − 1.56·33-s + 0.342·34-s − 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.124236111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.124236111\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.996823790011639034504087709035, −7.35860289940311835191352980006, −6.61711935264213320637953861850, −5.40373832339676397189556046868, −4.75726962661937589643355262311, −4.05861029634721041484966037455, −3.46965739296257293623385717043, −3.05271771719991476871708396650, −2.19096304790358757481743064780, −0.70239320449609775119628833386,
0.70239320449609775119628833386, 2.19096304790358757481743064780, 3.05271771719991476871708396650, 3.46965739296257293623385717043, 4.05861029634721041484966037455, 4.75726962661937589643355262311, 5.40373832339676397189556046868, 6.61711935264213320637953861850, 7.35860289940311835191352980006, 7.996823790011639034504087709035
