| L(s) = 1 | − 2-s + 3-s − 4-s − 6-s + 3·8-s + 9-s + 4·11-s − 12-s − 16-s − 2·17-s − 18-s − 4·19-s − 4·22-s + 3·24-s + 27-s − 2·29-s − 5·32-s + 4·33-s + 2·34-s − 36-s − 10·37-s + 4·38-s − 10·41-s − 4·43-s − 4·44-s + 8·47-s − 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.852·22-s + 0.612·24-s + 0.192·27-s − 0.371·29-s − 0.883·32-s + 0.696·33-s + 0.342·34-s − 1/6·36-s − 1.64·37-s + 0.648·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389787412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389787412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.54829543788129, −15.67760793329753, −15.15924464120678, −14.59122930274524, −13.97061073130418, −13.63168461932921, −12.96145751889923, −12.37528564052261, −11.68902764890869, −11.00152731462433, −10.30475672724044, −9.908406613822069, −9.086700852574739, −8.836899085764887, −8.330779704982344, −7.631393990594670, −6.820274632927206, −6.501097659061961, −5.316786989673608, −4.739135771892592, −3.818108723996313, −3.604810099796113, −2.208096203798496, −1.645507931283537, −0.5932000504268315,
0.5932000504268315, 1.645507931283537, 2.208096203798496, 3.604810099796113, 3.818108723996313, 4.739135771892592, 5.316786989673608, 6.501097659061961, 6.820274632927206, 7.631393990594670, 8.330779704982344, 8.836899085764887, 9.086700852574739, 9.908406613822069, 10.30475672724044, 11.00152731462433, 11.68902764890869, 12.37528564052261, 12.96145751889923, 13.63168461932921, 13.97061073130418, 14.59122930274524, 15.15924464120678, 15.67760793329753, 16.54829543788129
