| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s − 2·7-s − 4·10-s + 3·11-s + 4·13-s + 4·14-s − 4·16-s − 17-s − 19-s + 4·20-s − 6·22-s − 23-s − 25-s − 8·26-s − 4·28-s + 6·29-s − 8·31-s + 8·32-s + 2·34-s − 4·35-s − 2·37-s + 2·38-s + 4·43-s + 6·44-s + 2·46-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s − 0.755·7-s − 1.26·10-s + 0.904·11-s + 1.10·13-s + 1.06·14-s − 16-s − 0.242·17-s − 0.229·19-s + 0.894·20-s − 1.27·22-s − 0.208·23-s − 1/5·25-s − 1.56·26-s − 0.755·28-s + 1.11·29-s − 1.43·31-s + 1.41·32-s + 0.342·34-s − 0.676·35-s − 0.328·37-s + 0.324·38-s + 0.609·43-s + 0.904·44-s + 0.294·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.065763843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065763843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.019239640056801499451155011544, −6.95862088247859082470449924523, −6.55885676380523249064200876761, −6.04003574676563149948550306933, −5.12083385461365500721091569687, −4.06260458887764864200831508937, −3.35850330113448380256268152699, −2.19266743019243985366812613985, −1.58208827832733648095417043050, −0.64518823812616362205267109741,
0.64518823812616362205267109741, 1.58208827832733648095417043050, 2.19266743019243985366812613985, 3.35850330113448380256268152699, 4.06260458887764864200831508937, 5.12083385461365500721091569687, 6.04003574676563149948550306933, 6.55885676380523249064200876761, 6.95862088247859082470449924523, 8.019239640056801499451155011544
