| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 4·7-s + 8-s + 6·9-s + 11-s − 3·12-s + 6·13-s + 4·14-s + 16-s − 2·17-s + 6·18-s − 19-s − 12·21-s + 22-s − 6·23-s − 3·24-s − 5·25-s + 6·26-s − 9·27-s + 4·28-s + 32-s − 3·33-s − 2·34-s + 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 1.51·7-s + 0.353·8-s + 2·9-s + 0.301·11-s − 0.866·12-s + 1.66·13-s + 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.229·19-s − 2.61·21-s + 0.213·22-s − 1.25·23-s − 0.612·24-s − 25-s + 1.17·26-s − 1.73·27-s + 0.755·28-s + 0.176·32-s − 0.522·33-s − 0.342·34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966295491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966295491\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−9.549562360069387376237507614897, −8.304356927083541549554413106662, −7.64121883306403099205252345127, −6.51776113709287953207620827031, −5.96726521179698654958108725378, −5.40137673888935156754619083285, −4.34692530023923397895959701399, −4.03606380321009056145114520867, −2.01037870027126030361739144498, −1.03759100897086352742471870949,
1.03759100897086352742471870949, 2.01037870027126030361739144498, 4.03606380321009056145114520867, 4.34692530023923397895959701399, 5.40137673888935156754619083285, 5.96726521179698654958108725378, 6.51776113709287953207620827031, 7.64121883306403099205252345127, 8.304356927083541549554413106662, 9.549562360069387376237507614897
